To do someone's chores (or any sort of tedious work) for them; typically when the person is being too lazy or manipulative to do it for themselves.
A dialogue in which the term "to push someone's broom" is used::

George: "What happened between you and
Rosanna down in Texarcana?"

Sanger: "We're exs now. Wanted me to push her
broom."
by meluvcheese22 June 30, 2015
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