I typically see “bodged” used more like “hacked”: fixed/made to work in a clumsy way, jury-rigged.

So I’d botch my attempt to build a shelf by cutting a board too short; I’d then bodge it into shape by screwing on a piece of scrap lumber.

“Just repair it,” she hissed. “Please?”

“What, make a bodge job?” said the dwarf, his pipe clattering to the floor.

“Yes.”

“Patch it up, you mean? Betray my training by doing half a job?”

“Yes,” said Granny. Her pupils were two little black holes.

“Oh,” said the dwarf. “Right, then.”

-- Discworld

It's definitely a variation. I'm aware of people using botch like bodge but to me botch is specifically screwing something up, bodge is hacking something together.

Interesting, thanks, I was unaware of that distinction.