Axle to wheel bore ratio? This may be the wrong site but it's the closest one I thought someone would have knowledge of the question...

It's not for a bike but rather replacing the wheels on a power wheels... The axle is 7/16" can I use a wheel with 5/8" axle hole (bore?) or does it have to be tighter fit like 1/2"? Or even 7/16, 15/32?
Using an axle much smaller than the hole it fits into is sure to end badly.  However, you can make up the difference with a sleeve bushing.  Maybe you can find a standard size bushing that fits well, or maybe you'll need to invent one e.g. from copper plumbing pipe.  If you want to try the pipe approach, use 3/8" copper surrounded by 1/2" copper and you might just find a good enough fit.

You don't want the wheels to wobble...if they do, they will wear out rather quickly, and the vehicle's--ahem--performance may be adversely affected.