I'd use an electric sander and progressively sand it up to a high enough grit it's soft and the damage is as corrected as possible; I'd stop at 600 grit or so. The problem with particle board and MDF is it can be nearly impossible to fully repair liquid damage like this and thin wood leaves little room to repair it in many cases. Once you sand it back down (if it has a natural wood texture and it isn't contact paper), apply a similar stain if the wood is stained (you can even change it if you want) and then seal it in with furniture wax; just make sure you fully cover it so it's properly sealed and remember you need to redo this once in a while (but this also applies to ALL wood furniture once in a while so...). If the furniture uses contact paper (which is common for particle board/MDF furniture), sand it down, pick a contact paper you like, and refinish the top and sides of the desk.
On the other hand contact paper may be more tedious to apply well, but it will provide superior protection which will not be compromised by missed spots when waxing, but if it does wear down cleanup can be a process due to the adhesive in use.
The reason I mention the possibility of needing contact paper is a lot of particleboard and MDF furniture uses contact paper rather than natural finishing due to cost reasons, so you will most likely need contact paper to restore the finish on this type. Since it's a sticker you can either come as close as possible or pick a new color to cover the repairs.
You can see WHY I would probably recommend contact paper for MDF/particleboard here*, since all of that damage to the finish came with time and the only fix will be to sand it down, and then apply a roll of contact paper to it:
*The writing on the bottom is because I am reusing old furniture particleboard to make custom furniture; I'm refinishing it due to several years of paper damage, so I didn't feel bad about writing on it. The finish has to be covered anyway as the new wood patch will never match and I probably will not find a matched contact paper so I'm getting rid of all of it and putting a finish I want to put on it in the end.
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