It depends on how deep the cut is in my experience. A shallow cut or scrape usually just heals up. I don't really know about deeper cuts, a friend had a biopsy on a birthmark that healed with normal skin in the cut. I suppose tattoos might become distorted or loose colour in a deeper cut.

As Karlboll says- depends on a lot. Some types of skin are known to show scars etc. easier, the mark or tattoo or cut matters too in what happens or how visible any changes are.
In general if you get cut it will bleed most likely lol. That’s the same for a birthmark or a mole or tattoo or whatever skin.
The normal healing process will occur- in a general healthy and typical person the wound will close up (a deep or large cut may need sutures or stitches or such) and you’ll develop a scab.
Depending on the cut and your skin type you may get a scar. If you don’t get a scar the cut will generally heal and look the same as before- EXCEPT-
If you have a tattoo and the cut doesn’t go as deep as the ink the tattoo will not change at all. If your tattoo wasn’t inked deep or is maybe very new and hasn’t “settled in” the deeper layers of skin, the ink may be removed or damaged by the healing and your tattoo can end up with missing parts of the design or distortion. Missing ink can…

.. generally be fixed with touch up work while distorted design may or may not be easily corrected and you may require removal or a “cover up” done if you don’t want any visible distortion. For birth marks, it depends as that’s a very general term. So, darker patches of skin are often caused by various factors that cause those skin cells to produce more pigment. Often times when such a patch is cut on a person who is naturally lighter colored, the initial healing may be that lighter color, but where the skin cells are being signaled such as in their DNA “instructions” to darken, they will simply darken again with time. Usually to prevent that, the entire area needs to be removed so that you don’t have any cells duplicated which pass the instructions to carry the darker pigment. This is the case such as with mole removal where if you do not remove the “whole” mole, the mole will simply return.

So if you remove some but not all the skin such as a cut or scrape on an area of darker pigment, it will probably heal back in and be darker. The shape or size may change. It is rare but possibly for raised marks to become flat after healing but unlikely. It is more likely but not particularly common that flat birthmarks or pigment variant areas may become raised after an injury removes a deal of skin- though a cut isn’t likely to trigger this unless it removes a good portion of the area. In general we can say that without scarring the topography of the skin may change in such an area.

If you get a scar- healing again will generally follow the same process for scarring as any other area. With a tattoo or birthmark you will now have a scar at that location. Scar coloration, shape, and topography vary. On a birthmark the scar may be darker or lighter than their genera skin tone- another saves tend to be as all. If your cut gives you a scar where you have a tattoo the design will likely be compromised where the scar is and perhaps a bit in the surrounding area. The rear depends. Scars can be “raised” or “sunken” etc. the severity of the wound and size of the scar can impact what happens. In some cases you will still see the ink in th scarred portion of skin, though lines etc. may look wavy etc. because the skin is no longer flat. In other cases the ink won’t be visible, and in some cases you will see perhaps some of the ink in places which can be distorted like the above examples but in other places see no ink.
It is possible to tattoo over many types of scars etc.

So it can be possible to “fill in” a tattoo if you get a scar on top of the skin. It can be difficult to do well since the skin may not be “flat” in the scar area and the artist might have to make the design so that if it were laid flat it would look bad but because your skin isn’t flat it appears “correct” as though on “flat” skin. An optical illusion basically. Because the scar area may be a different color or density of skin and the topography issues- matching the colors of the tattoo over the scar can be a challenge. If the tattoo is older, you may also need to try to match the older ink which can also add challenge. Depending on the design it may be costly and there may be some pain or discomfort, but when doing work like that it might be best to get the whole tattoo touched up. Tattoos tend to “fade” as they age, colors or shades like black become less vivid and fine details can be lost or become blurry etc. for that reason it is common for people to have older tattoos touched up

Basically you have an artist “trace over” your old tattoo with fresh ink, they color in colored portions and redraw over the lines. This allows fresh vibrant ink to show through and the artist can make corrections or changes such as filling in details that have become blurry or adding certain things that maybe weren’t in the original design if you like. Depending on the size and location of your scar, the complexity of the tattoo design, and the age/appearance of the tattoo- the best results might make getting the whole tattoo done in fresh ink the best plan so that you don’t have one area that is obviously newer or sharper than the others or to mitigate some of the challenges in blending the new ink and new tattoo work with the old.

There are other types of birthmark besides discoloration and cuts or scars on those vary as well- the general principles liked out here apply. You may have topography changes and depending on the circumstances the size or shape of the mark may change as well as some coloration changes being possible.
The scar or healing of the skin may be impacted by the type of birthmark and underlying reason- if the birthmark is caused by something that effects how your new skin cells will behave then the healed results will likely be effected. It is often the case with many types of birthmark that even without scarring you will see a permanent difference in the healed area. How noticeable this is depends largely on the specifics of the cut like its size. This is because birthmarks are often underpinned by variance in the “instructions” to that area of skin on how it is supposed to be.

So for instance if you have a birthmark that looks like little “waves” on an ocean because the skin has instructions to form in raised patches, before the injury your “waves” may have formed a pattern that was even or looked perhaps almost geometrical- symmetry and such. After healing you might instead have an area where the “waves” are higher or lower or wider or a different shape or even “pointing” a different direction- the formation of a specific pattern can be effected by things are variable. As we age the collagen and other things in our skin can change. Our nutritional levels and the rate we heal at can change, with how fast new tissue forms potentially impacting the end results. The moisture in our skin or the air can change things and all manner of factors such as movement which stretches or moves skin as it heals or pressures such as interacting with our environment or from dressings and bandages etc.

So overall there are a lot of things that can happen and human healing can be unpredictable, but some general things are likely to happen.
Also please note- not that you are considering this etc, but I must mention- any intentional attempts to remove birthmarks or moles etc. at home are generally not good ideas. Often times the mark returns and even when a medical professional attempts to remove such marks they can return. They can also return more severe or “worse” in appearance than they initially were, and there is generally always a risk of scaring when removing marks. It is also very true that unskilled persons attempting to remove marks may cause injury- life threatening would be somewhat difficult or rare, but damage to skin and underlying structures or even nerves is possible as well as loss of mobility if certain areas of skin that routinely flex or stretch are damaged and even nerve damage and potentially serious side effects. Infection is perhaps an even greater danger and

could actually be realistically life threatening or at least threaten a limb etc.
accidents happen and we get cut or scarred sometimes- but most of the time even with a tattoo or birthmark you’ll heal just fine and barely if at all be able to tell there was ever an injury. With tattoos it is usually possible to keep your tattoo or a new or modified tattoo even with scarring. I also want to mention that scars are nothing to be ashamed of and can be reminders or even fond memories- and many can look quite cool or bring a level of interest much the same as body art or piercings, so nothing wrong with scars.
This was a good question and something that was fun to answer. I don’t know if my long answer was fun to read but hopefully it was at least informative or helpful in satisfying your curiosity!

Thank you that was nice reading before bed :) I learnt a lot