field mice eating my seeds/seedlings

It is spring down here and I've got my seeds starting in the trays. I thought it was birds that got my squash/pumpkin/cuke seeds but just cleaned out the area and found a whole load of seed husks and field mouse poo.
The smallest metal mesh I can get here is just over 1 cm hardware cloth, which I think is probably too large for field mice, but I made cage-type things for my trays anyway. I put down mousetraps in the area where I leave the trays at night.
(when attacked, the trays were on top of my mouse-obsessed dog's kennel. He and I will be having a talk later.)
We have a few feral cats who seem to crap in the garden when it's convenient for them but haven't put in an appearance as mouse hunters. I put down "distraction food" for the mice and birds (buggy corn and some old bread that the rabbits didn't want to eat) but the mice don't seem to care.

Any other ideas that will keep my seeds safe, I'd gladly welcome.

Maybe your dog has a good idea? It sounds like he'd be a better mouser than the cats. I don't know if there's a way to bait the kennel to attract mice there? But yeah as a former mouse breeder, 1cm mesh is not going to protect anything from mice. I don't know a good short-term solution other than unleashing rat terriers on them, but long-term I'd be encouraging snake habitat and owl habitat. Mice have a good sense of smell and are going to make a beeline for whatever's tastiest, and feeding them may help them reproduce in larger quantities.

Now I'm not sure how you're starting the seeds, are the seeds themselves exposed on the trays or are they in some kind of substrate? If you want to sow them directly you can clay ball them, otherwise it might be safer to start them indoors.

Ugh Sarah, that's what I was afraid of. I can't even get metal screen, only plastic! I think tonight I'll put the one tray with the new starts on top of a plastic trash bin, i know they can probably scale that too, but why make it easier. Seeds inside means they would take up the entire kitchen (or my office? eep), I suppose next year I may do that if it comes to it.
I live on a tiny urban farm (my plot is about 7 m by 20m), so there are limits in terms of what I can do to encourage habitat. I know we have owls and hawks here in town, they eat the pigeons.
The dog has caught mice before but apparently he doesn't care if they're partying on top of his kennel (he's not locked in, it's just for him to sleep in should he choose to do so). I suspect he doesn't see too well at night, and would rather be barking at the street dogs and the garbage truck than protecting my seeds!

I was wondering if maybe the mice are going for the seedlings for not only food but hydration as well.  Maybe some water left out for them would keep them off the seedlings???  I had a cottontail and a few birds coming into my greenhouse earlier this year, and I left a bowl of water for them so they wouldn't eat my plants looking for hydration.  I also had a cover crop and the cottontail went ahead and trimmed it all for me and converted it into manure as my young plants grew up through/with the cover crops.  I've seen a few cherry tomatoes here and there that the mice were getting, but not enough for concern.

Another thing I was thinking...maybe grow them a tray of microgreens?  

It's kinda hard to want to do those things though because they'll keep coming back and soon you'll have many more mice.  But just some thoughts I had.

it has been dry, and certainly the birds will go after moist soil even if mice won't. good idea.
I was thinking more about the setup-- our lot is so narrow that the house reaches the property lines (to get from the front to the back yard, you need to go through the house). When the trays are in the back of the house, the mice seem to have enough to keep them busy (between the corn I threw out for them, the rabbits and their mess, the birdbath, and the compost pile). I had the trays in the front, where there are none of these "amenities" but figured the dog would be enough to keep the mice away. I suppose the mice in the front yard are hard up for snacks. I'll keep the seedlings in the back, and maybe toss some birdseed in the front yard.
Thanks Joshua and Sarah!