- My cousin had an internal cyst that burst yesterday.
- My cousin who had an internal cyst had it burst yesterday.
- My cousin with an internal cyst had it burst yesterday.
Are the above sentences acceptable?
Hello, Ahmad. Welcome back. I agree with Gustavo's answer. Assuming it isn't important to you to indicate grammatically, with restrictive modification, that you have more than one cousin, I recommend:
4. My cousin had an internal cyst burst yesterday.
That sentence would not naturally be interpreted as indicating that the speaker has only one cousin, nor as indicating that s/he had a doctor burst the cyst for him/her. It is naturally interpreted as indicating that s/he already had a cyst.
Regarding (2), which I find very awkward, it is considered grammatically questionable to modify a possessive noun phrase with a restrictive relative clause. If the restrictive relative clause is important to you, I recommend:
2a. The cousin of mine who had an internal cyst had it burst yesterday.