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.
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