You'll need to remove the hinge first, but whist you do that it might be worth taking both of them off removing the door. You'll need to fill and repair the damaged portion before being able to install the hinge again.
Sand the flaking chipboard and laminate with 180 grit sandpaper, being careful not to damage the surrounding laminate. Fill the holes with Selleys Liquid Nails Clear Construction Adhesive, then let it cure fully. Apply Turbo Builders Bog over the damaged area waiting for it to cure and sand flat once again. You can then paint the area a similar colour to the cupboard disguising the repair. Once this process is complete, re-drill the fixing holes and install the hinges into the solid material.
We do stock that type of hinge and if you take it into your nearest store they will be able to match it up for you. However, from the picture the hinge doesn't look damaged it just appears the chipboard is so disintegrated it won't hold screws anymore.
If you need further advice on the installation of the hinges or have any questions, please let me know.