I get this too, I blame it on a slightly bent chainring/spider. To me it sounds vaguely like one of the clicky ticky pingy noises you hear when you're changing the front gear on a derailleur bike, so my guess is the noise results from the chain being pulled sideways (by bad chainline) off of a tooth right as it is clearing the chainring (uninformed opinion.)

I need a new chainring and/or cranks. For now, I just make sure there's as little slack as possible.

Replace your chain, perhaps?

Sometimes stuff gets on and your chain gets messed. It doesn't flex properly - that tends to make that "click" sound like you mentioned when you're accelerating. Had a similar problem on my road bike. Replace the chain and see if it works. Or borrow a friend's chain and see if that works out for ya if you're short on money and don't want to get a new chain.