I have had letters delivered to me that had abuses.
I had letters delivered to me that had abuses.
The second is simple past use of had. It means that at some particular time in the past, such letters were delivered to you.
The first is the slightly more complicated present perfect. It means that at undefined time in past, such letters were delivered to you.
We can examine this further by adding specific times:
*I have had letters delivered to me last Sunday
I had letters delivered to me last Sunday
I have had letters delivered to me in the last week.
The first (marked with an asterisk) is incorrect because it defines a specific time, and the form does not work with a specific time.
It is used to describe experiences one has had in the past (and that hence influence the experience with which you speak today), changes over time, uncompleted actions, and things that have happened more than once.
By the way, abuse as a countable noun in terms of insult is unusual. It would be more common to say "…that were abusive" or "…that contained abuse".