Can I use AppleCare to fix a cracked screen one week after purchase?

If you are referring to normal AppleCare, then it does not cover accidental damage and you will be paying full price for repair. If you are referring to AppleCare+, then the coverage begins as soon as you purchased it and covers accidental damage and you will pay a reduced price for repair. Make a reservation at your nearest Apple Store's Genius Bar or Apple Authorized Service Provider for a screen replacement.

Genius Bar Reservation and Apple Support Options - Apple

AppleCare does not cover physical damage to the phone.

If you purchased AppleCare plus, you can pay a fee and have it fixed.

Contact Apple support for your options.

Can I use AppleCare to fix a cracked screen one week after purchase?

Yes and No

Yes, if you purchased a separate AppleCare+ policy......before.....the screen was cracked.

No, otherwise.