Well I think you’re operating on a belief that “suffering” and “mercy” are some sort of opposites and I don’t think that’s necessarily true.
Obviously, you can apply mercy TO suffering and lessen or alleviate it.
But I think mercy can be felt in other ways, and that suffering can be felt despite mercy. And sometimes in spite of mercy.
It’s a complicated situation and I’m not a theologian. I misunderstood the basis of your OP because I didn’t realize that when you asked about mercy that you were asking a broader question about why suffering is allowed. I can tell you how God has applied his mercy to me (to the extent that I’m aware of it) but I am not equipped to explain suffering.
I really do suggest you start with “The Case for Faith” and go from there. Again, there’s entire books on why suffering is allowed.
One thing I’ve considered when contemplating suffering myself is that the big picture for God is obviously much different. For example, if you’re a parent and you’re teaching your child to ride a bike and the child falls and cuts their knee, that’s horrible for the child at the time. Why did that happen? How could you let it? Why can’t you make the pain go away? Why are you putting medicine on the cut that makes it hurt worse??? But you know, as an adult, that this is very temporary. A child will skin their knee sometimes, but it’s going to be okay. The child, from their perspective, doesn’t yet understand. I think sometimes that’s how it is- but on a much larger scale- on our view of suffering vs. God’s view of suffering. This life we have here, it’s a blink of an eye compared to eternity.