- 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.
All three sentences are fine. In (1), the speaker might have only one cousin. In (2) and (3), it is clear that he/she has more than one cousin, one of whom had an internal cyst.
Also, in (1) it is clear that the cyst burst spontaneously, without any human intervention. In (2) and (3), it might be the case that the speaker's cousin had somebody burst the cyst for him/her.