If you are sent on an errand or run an errand for someone you are doing a small job for them that takes you away from the home or where you would normally be. [A short trip taken to perform a specified task, usually for another].
Delivering a message to your mothers friend down the road or going out to collect the groceries would be 'running an errand'.
"To be" is not an idiom - it's a verb.
Pest is not an idiom. It's a word.
The idiom "apple shiner" means the teacher's pet.
The meaning of the idiom in the pink of health means being in good health.
The idiom means impress someone is egg on
If you "send someone for errands" you're giving them a meaningless task to do just to get them out of your way.
The expression is not idiomatic. It means exactly what it says. To be sent on ( or for) errands means to be out on a shopping trip, or such like, for someone. Mother sent me on errands to the grocery store and the dry cleaners.
The messenger and/or servant
"To be" is not an idiom - it's a verb.
The carriers were sent on errands.
Pest is not an idiom. It's a word.
The idiom "apple shiner" means the teacher's pet.
The meaning of the idiom in the pink of health means being in good health.
The idiom means impress someone is egg on
It's not an idiom - to cope means to deal with, or to handle
"Old hand" is an idiom meaning having lots of experience.
It is not an idiom. It is an expression. The difference is that an idiom's meaning cannot be derived from the meaning of its individual words. In the expression wolfing down food, the meaning is clearly derived from the meaning of the words, and people have been saying it for hundreds of years.