Can apple care plus repair a heavily damaged phone?

Only Apple can say. Take your phone to them and see what they say.

Go to this page to find your nearest Apple Authorized Service Provider (AASP) or Apple Authorized Distributor (AAD) --> Find Locations

How to make a Genius Bar appointment

1. Go to Genius Bar Reservation and Apple Support Options - Apple

2. Go to "Get hardware help".

3. Select your hardware and continue selecting options requiring replacing parts until you see "Bring in for Repair"

4. Sign in with your Apple ID

or:

Refer to this document for ways to contact Apple ➞ Choose your country or region - Official Apple Support

Select your country (also look for "other" regions), then a product. If you don't see one that handles your issue then keep experimenting with selections until you reach one that gets you a chat session or a telephone call and get the representative to redirect you.

or:

Contact Apple for support and service by telephone ➞ Contact Apple for support and service - Apple Support

Then there is your answer.