I'm not familiar with mesquite sap, but a lot of sap can be removed with a basic washing. For the stubborn spots, straight car wash soap placed on it and let to sit for a few minutes helps break it up. This is also very safe for the paint in comparison to solvents. But, some spots will be too much for car wash concentrate, so solvents become necessary. WD-40 is a gentle cleaner/solvent, and may loosen it if allowed a few minutes dwell time (it is excellent for tar, too). 3M's Adhesive and Wax Remover is good stuff. But it is strong, and will strip everything off your paint (wax, sealants, polishing oils, etc; but is safe for cured automotive paint). There are also a multitude of products on the shelf for bug, tar, and sap removal. But I think most employ a type of solvent similar to the 3M stuff, and the 3M stuff is extremely versatile for other household cleaning chores. If you use any type of solvent, I'd advise you to use a wax or sealant over the area when done.
Other people have had good luck with Goo Gone. I haven't tried it for sap, but it is reportedly easier on the paint than a full blown solvent such as 3M's Wax and Adhesive Remover. It may be worth a try, and would have other uses around the house. Personally, I try to buy products with more than one use, as my detailing collection has grown quite large with specialty items.
Another option would be to freeze it, then "chip" off the pieces gently. Similar to removing bubble gum or candle wax from surfaces like fabrics. Works well, and may help with big blobs of sap (I couldn't tell from your post if you have several big blobs, or more of a fine misting). To freeze it, a can of compressed air (dusting air for computers) sprayed upside down on the spot will freeze it pretty much instantly. Anyways, it could be worth a try. Just be gentle when picking the sap off the surface so as not to mar the paint.
Hope this helps.
Dave