Batter replacement

Apple will replace the battery but they will require the cracked screen to be repaired first. If your device is outside of it's limited warranty then it may be more cost efficient to have your battery replaced at a third party phone repair location.

Apple may not touch it due to the damage could get worse while trying to open the phone.

But the only way to find out is to make an appointment and see what they say.

Genius Bar Reservation and Apple Support Options - Apple

That will be a decision for the Apple Service shop.

Make an appointment at the Apple Store or take the phone in to an authorized Apple Service shop. You can use the link below to find authorized service shops in your area.

Find Locations

Cost of a replacement battery is $69-89 (US) depending on the model of phone that you have.