There are several methods to do this. One is to (as Steve said) patch the missing section, but instead of using 1/8" or 1/4" plywood use Veneer, you can get 10mil and 30mil 4x8 sheets in a roll...it will be expensive but the job will turn out better. Contact your local Hardwood supplier or one of the mail order woodworking companys, or a local cabinet shop/woodwright.
If you want to try to remove the damaged veneer you can take a damp rag and an iron, lay the damp rag on the desk and press the hot iron over it moving and rewetting from time to time . This will take some time as you are steaming the top off. Use a wide putty knife to get between the veneer and substrate. Be very careful as you can easily get burned by the steam. Some times this works and sometimes it dosent.