ok welcome to the site
I am guessing you mean a WARN brand winch?
and hand warmers?
and NO if motor is running your charging system might NOT have enough power to run either, all the more so the winch, it draws a LOT of amp's to make it work
and since you didn;t list what ATV you have
most atv's have pretty low powered charging systems, as they were never really made to run SO many added on things , most guys add, your BATTERY is what supply's MOST of that power, so a BAD battery will cause this
also, could have a blow fuse to each item too?
, a broken wire, or bad connections, , hand warmers are prone to go bad in time too, they burn up, so to speak? and are more critical to abuse and failures from that, or age alone
I suggest getting a GOOD battery in play here and see what happens from there