What can I do about the mud in my backyard as I don't have much grass?

Install turf (synthetic grass), mulch, put a pond in (the puppies will like some fresh water to sip) or gravel.

go to local feed store that sells horse/farm supplies or online and buy bales of cedar shavings used in stalls and put it in/thru out muddy areas(you'll have to see how many bags you'll need/how big area is) Cedar also repels bugs & smells good.you can get/use pine shavings it is cheaper maybe mix the two types. You will need to add more later or put down straw first so it doesn't get trampled into soil too easily. just depends how much you will need, mixing the 2-shavings and straw will be cheaper.Do NOT add lime it will burn their feet.gravel will cut their feet unless you use pea gravel or rubber mulch (very costly) took me a while but found this video of some one who used it (we used it at our barn and in areas to control mud after heavy rain/snowfall as well as stall bedding) https://www.youtube.com/watch?v=MCCIabwd2l4

It sounds like you need to drain your back yard. We had that problem and we put in French drains to let the water drain off when it rained so often in our location. They are a lot of work but not difficult to install. You just have to make sure the end of the drain away from the yard is a little lower than where you start in the yard. Check with your local home improvement center about getting them. We dug a hole about 8 inches deep and a foot wide, and put in gravel, the French drain, then covered with gravel, then put back the dirt as much as we could (we had excess and put it in our flower beds or vegetable gardens). It worked perfectly and solved our muddy situation. If you live in the city limits be sure to check with your local utilities about digging etc

I was looking into this. I don't know if there's a dealer in your area, but it's worth checking into.

https://k9grass.com/

You could grow grass I have three to keep planting it if they dig holes put poop in the hole they won’t dig keep doing that they will stop digging takes a wile but they stop