Half lap joint advise needed

I've never been one to be afraid of admitting my mistakes and lack of woodworking skills so I come to you with a question.
I'm building a cart for my newest street organ and I'm working on the handle that I will attach to it. I have attempted my first half lap joint. (You already know what I'm going to ask don't you). Anyway, I used my table saw and I knew that finding the exact 1/2 measurement for each side of the cut was going to be difficult so I tried to creep up on the depth slowly however the last cut was about 1/16" too deep on each piece and now my joint is about 1/8th" too thin. So my question is...can I cut a thin spacer to glue inside of this joint or would that completely ruin the strength of the joint? I should have taken more time to find the exact middle of the wood. However is there a way to save these parts?

You could add a thin piece to both sides of the joint and then recut to dead center. I do not believe the joint strength will be compromised. Just make sure the two bits are flat and clamped tight. You should be OK.

only thing i would add to rennies thoughts is to make sure you run the splice the same way as one of the pieces that are to small.. you could make it with one piece added

ok, thanks guys. I went out and cut a thin slice about (3/32") of the same wood (cherry) and glued it in between the two halves with the grain running the same direction and clamped it all together. Made sure I had glue spread well and joints were tight. I'll let it set up overnight and see how it looks tomorrow.
I'll not make this mistake again. Lesson learned.