Patch and Paint a Desk Surface
I would use a high-performance 2-part wood filler (it's kinda like auto body filler). Overfill the area then shave it down with a Surform plane (a cheesegrater plane), then use coarse, medium, and fine sandpaper to smooth. 

That desk could present problems regarding paint, the surface (it looks like a shiny laminate) is designed for stuff not to stick to it. I would fine sand the top, paint it with a "difficult surface" primer like BIN, then finish off with a high quality gloss paint. If you use lame brushes, your paint job will be lame so do the opposite of that.