This is a problem I find with the wild turkeys, scratching the mulches from where I want them to the grass where I do not want them. I have seen many smaller birds, Rose Breasted Grosbecks, Cardinals, Blue Jays, etc. scratching in the mulch but they are not kicking the mulch out of the planting bed into the grass. To keep the larger birds out of these planting beds they are fenced, but that fencing will not keep the smaller, much less destructive birds out, and I do not see a problem with them.

I wouldn't worry about it. It shouldn't be a big deal. Think of the birds as providing some air flow if you want a positive way to look at it. Just put in some new mulch on the bare areas once in a while. Since I tend to have wood chips in abundance I use those more than leaves and critters don't knock the wood chips about so much.

If you are concerned about weed suppression, then yes, your scattered mulch won't perform as well, but if you're sort of casual about it and you like feeding birds, it won't hurt anything.

If it does become a problem for you, you might try putting a layer of cardboard under your mulch. I doubt that would discourage my chickens for long, but smaller birds might give up more quickly.

I put cardboard on top of my potted stuff in the summer to discourage squirrels and it works great. I cover it with something, of course.