Is it possible to...

Yeah. Plant a ton of flowers in the dirt patches, water them daily if you can reach them without entering the dirt, never walk in the dirt spots, and, if you can, TTing a month or so ahead and then going back usually removes all the grass wear.
 
Alternatively if you don't TT (like me), I've found simply leaving it gets the best results. I planted flowers all over a dirt patch but found it was getting worse due to having to walk near it daily (where I was walking was obviously causing more wear). Since I removed flowers and just left it alone, I noticed improvements in a few days and it gets better daily. I don't even have Beautiful Town ordinance either.
 
Alternatively if you don't TT (like me), I've found simply leaving it gets the best results. I planted flowers all over a dirt patch but found it was getting worse due to having to walk near it daily (where I was walking was obviously causing more wear). Since I removed flowers and just left it alone, I noticed improvements in a few days and it gets better daily. I don't even have Beautiful Town ordinance either.

Pretty much this. If at all possible, don't run or even walk on the damaged grass. Any form of interactivity between you and the damaged area registers in the game as grass wear, preventing the grass from growing back. So even if it means taking a more inconvenient route to get around, do so until the grass is back to normal.
 
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