Actually it's possible to have a desert at the equator. It's rare but if the winds are in the right direction, it's possible as long as there is little rain. If mountains are blocking the winds like in South America and Central Asia. If it's too cold it becomes a cold desert.
It may be possible (I've never heard of a 2000-mile long equatorial desert that stretches 20 degrees north and south of the equator) but I don't think his land allows it anyway. The equator is almost constantly the hottest place on a planet and the geography would have to be extraordinarily influential to stop it. His geography doesn't seem to merit such a difference.

I'm not sure why you say that he should move the equator. I agree that the western part is probably too dry (but since I don't have the info I can't tell exactly) but I think it's simpler to just change the climates for that area since the rest looks ok. Except fort the Laurentian climate...
Since he has human locations marked on the map, I went with the assumption that he had done some cultural work on the world. In my experience, moving the equator south so that the climates make more sense has almost no influence on the cultural work he may have done up to this point, while changing a bunch of climates would mean that many civilisations would be subject to a redesign, since climate is such a huge influential factor in human development. I was only trying to suggest a change which would have required the least amount of work.