I think I may have sprained or broken my wrist. I'd like to have it X-rayed, please.

You have had an accident and have gone to A&E - the accidents and emergencies department - at the hospital.

Sprained. A sprained wrist. (Adjective.) I've sprained my ankle. (Verb.) It's also a noun - I have a bad sprain.

To sprain a joint - wrists and ankles are the common ones - it means to damage the joint or the tendons by violently bending the wrist / ankle etc. the wrong way. Very painful.

I may have broken - may means possibly. I have possibly broken my wrist.

When we use modal verbs (like may) in the past tense, the structure is: may / must etc. + present perfect.

I may + have + sprained it.

I'd like to have it X-rayed. Could someone X-ray it please?

To have something done (to have + something + past participle) is a passive structure.

We could say 'I need to have my eyes checked,' for example, which means 'I need to find an optician who will check my eyes.'

This sentence is an example of modal verbs, may, modal verbs + present perfect, medical.