Wouldn't be hard to adapt Woody Guthrie's Riding in my Car to meet the situation:

Brrm, brm brm...(mower noises to tune)

Take me riding on the mower(2x)
Take you riding on the mower
I'll take you riding on my mower

Climb climb, rattle on the big seat
Spree I spraddle on the big seat
Turn my key, step on the starter/ TAKE....

And so forth.