'75 FXE with Bendix carb. Occasionally it backfires though the carb when I give it a touch of gas to accelerate from a stop. Frequently the motor dies after the backfire, but starts right back up with a kick.

Timing?

 

its lean -- the acc pump is skipping a beat at the start of the spray usually

sometimes its the low speed jet is starting to clog over << gas we have now

But don't for get to adjust the mix screw after you fix it and leave it a taste rich

 

Lean as John says and if adjusting doesn't do it check for an intake leak at the manifold.