1
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Phrase(s): run an errand [and] do an errand; go on an errand
to take a short trip to do a specific thing; to complete an errand. • I’ve got to run an errand. I’ll be back in a minute. • John has gone on an errand. He’ll be back shortly.
McGrawhill's American Idioms And Phrasal Verbs
2
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Go to perform a commission, as in I spent the morning running household errands? to the cleaners, the supermarket, the hardware store. [c. 1500]
American Heritage Idioms