I am not a native speaker. My research suggests that "had/has set off on an errand" is grammatically okay, but it's not idiomatic.
The only examples I found are:
- "Wildwood Road" By Christopher Golden
He was still holding her that way when the sound of an engine cut the night. Jillian had set off on an errand ... a short drive to that elegantly antique gas station, so well preserved, an echo from another era.
- Bernard Shandon Rodey
The defendant had set off on an errand to deposit his employer’s money at
the First National Bank, but had ended up gambling most of it away at “Hope’s corner faro bank” instead.
- Shamballa Part 3: The Philosopher's Stone (A fan fiction)
My escort, Sergeant Beregond, has set off on an errand I appointed him. I already gave him instructions to meet me here.