There are 2 basic solutions, but the general/assumed fault lies with tires mounted on painted rims.
Won't the foam get mashed down with a heavy weight especially with a had truck or wheelbarrow? As far as the fix for the mower tire I'm with havasu. I'd put a tub in it. That's what I did with my wheelbarrow. Otherwise I'd take it off the mower and air it up and put it in water and find the leak. It could just be the valve stem leaking. Might just need to tighten the core. If its the bead, peal he tire off and clean it up and put it back on. I wouldn't take it to a shop. Small tires like that aren't that hard to take off the rim.
I saw it on the internet so it must be true. I've never tried it. But I'd think on a wheelbarrow or hand truck it would be OK. You can buy wheels for both that are essentially foam filled. They aren't solid rubber. https://www.harborfreight.com/13-in-flat-free-tire-with-steel-hub-61606.html
My friend had a scrap business and was always driving his skid steer over sharp metal and going thru tires all the time. He also used it for snow removal. He took his tires down to a place and had them filled with some kind of poly. It reminded me of what super-balls were made of as a kid. The tires were super heavy but that kind of helped with snow removal and wasn’t a problem in his junk yard. He came up to help me plow out a old friend of ours place and I looked over and his tread was totally gone and he was driving on the poly fill material. The side walls were still there and some of the thread had inverted and was sticking out to the sides. He told me he needed new tires but he never had better traction so he was going to wait till spring.
