Well, you can add operators, though you almost certainly should think about more idiomatic solutions first, it's entirely possible to write operators to allow a statement like:

  arr .~ (i,j) .= arr ! (a,b) + arr ! (x,y)

You make orphan instances for Indexed and Num, and then .~ is just a slight tweak on the adjust function that Indexed provides and .= is literally a direct synonym for $ that just looks better in this context. This is actually more flexible than most languages, because now (arr .~ (i,j)) or (.~ (i,j)) can be named and applied to multiple things should that be desirable for some reason. Also note that this code is very weird, as arr is not an array, but an IO action describing how to produce one. Operations on it actually produce diffing instructions to be applied elsewhere. I have also not tested the performance.

These exact things are not in base because they are discouraged and not supposed to be easy. Note that the lens library provides operators more or less just like this for a wide variety of data types, in a safe and composable way.

Taking IO actions just to do the bind for the caller goes against the benefits you usually get from purity.

OK, I know this isn't nice but it's already part of the way there

    writeArray arr (i,j) =<< ((+) <$> readArray arr (i,j-1) <*> readArray arr (i-1,j))

This is not only not nice, it's just a bad idea. (and I understand it!)

I think you may have misinterpreted my comment.