Often soffits like those do carry central air and heating and thus are expensive to remove other times they are to fill the void between the cabinets and ceiling for easier cleaning than it is easy to change out. You would need to know why yours are there. Hopefully, and it looks like it was, for cosmetic not structural reasons only

Unzip the drywall and take a peek first.

Unzip? I'm assuming is make a small hole.

Some times they have duct work in them or electrical. Of course you will have electrical wire for the light above the sink. Some times there is nothing in them. They are not structural so you can take them down. The only way to tell is to grab a hammer and smash some holes along the soffit and take a peek. Sometimes you can just palpitate along the drywall. If there is a distance greater than a two by four that doesn't sound hollow it is probable duct work.

Well there is definitely electric in there because you have a light above your sink. Easily removed and usually wiring is easily rerouted. You probably are going to remove popcorn ceiling anyway so might as well go all the way.

Have ceiling tested if house built in the 70's for asbestos.

If there is attic above, it won't be as big a deal to reroute any surprises. It looks like there is not ventilation over the range but the cabinet is placed as though there is. I wonder if a hood was removed and the duct covered when the cabinets were painted.

By "unzip", I mean cutting and removing an 8"x8" or so square. If you see mechanicals that can't be cost effectively removed, you can mud and tape the piece back in inconspicuously.