Put the new stuff on top of the old.

Here's how it goes.
Pull the old stuff back a bit, scratch the fertilizer lightly into the soil around the roses, replace the old mulch, then top it off with new/fresh stuff.

Nutrient sources like granular fertilizers leach right through mulch, there is no need pull mulch back, scratch them in etc.

And the bushes are better off if you don't dig among their roots.

If this material is instead an organic top dressing, the soil system occupying the undisturbed mulch should likewise be able to integrate it and make it available to the plants.

Nobody pulls back mulch, scratches in fertilizer and the like in nature. Mulch is the garden version of the litter layer in the woods - where plant and animal debris falls to the ground, gets taken into the soil system without human involvement.

I was surprised by a garbage man on how often mulch is cleaned up and replaced with new. He said that he handled more bark than I did as a landscaper.
Maybe the public needs to be educated on this.
I have a friend whose wife doesn't like the look of wood chips, but can't afford a full load of ground bark. I suggested he put down wood chips first and then cover them with a cosmetic coating of bark. He did, and now his wife is happy and he has a decent thickness of mulch to do the job.

That's what I did when we were in southern Oregon for 3 years.

Had several companies and the utility company line clearing crew deliver many loads of chips. After they were spread and allowed to flatten for a few weeks, I covered with bark mulch.

Got a few mushrooms, but not many. I find the mushrooms interesting actually.

My preference is to avoid getting hardwood when the seeds are on the trees. Maple for example, can have a lot of winged samaras.

M. D. Vaden of Oregon

One of the reasons to use wood chips is to avoid the drawbacks of bark.