Should I reinforce my lap-joint?
Use waterproof glue, make the joint tight-fitting, and leave the screws on the shelf. A lap joint is one of the strongest of all the glued wood joints, and a well-fitted glued lap joint will be at least as strong without the screws as with them.

But... there's a condition. How wide are the two planks forming that joint? And which wood is being used? I ask because of wood's inherent expansion/contraction with humidity/moisture changes. If the joint is too wide for that species of wood, and the wood isn't extremely well sealed against moisture intrusion, then expansion/contraction will destroy the joint; nearly all woods expand &amp; contract across the grain far more than they do along the grain. As the wood becomes moist, it'll become wider but not longer.

Certain woods have a very low expansion/contraction rate. Other woods have a very high expansion/contraction rate.

If the wood is extremely well sealed, its inherent expansion/contraction will be minimized, but can never be totally eliminated (there's no such thing as a perfect seal).

If you're using wide boards, and that species of wood is subject to high expansion/contraction, then neither glue nor screws will prevent the joint from failing. Better in such a case to make a different type of joint, like a dovetailed saddle joint pinned across the "inside" corner, for example. That's much more difficult to make, though. Another option would be a long mortise with a tenon narrower than the length of the mortise.

Good outdoor gate joints are difficult things to make, since they're exposed to the weather and tend to collect pockets of water after a rain.