To inflate a tire, there must be an airtight seal on the bead between the tire and rim. When the seal breaks on a small tire, like the one on a wheelbarrow, it is impossible to inflate.
To reseat the bead, attach a ratchet strap around the tire and slowly snug it up until the bead pushes out to the rim:
After the ratchet strap applies enough pressure, the bead will seal against the rim. Attempt to air-up the tire:
Don’t put in too much air with the ratchet strap attached. Releasing it while under too much pressure may make the strap snap loose and take a bite out of your finger:
Fully inflated tire: