weigh with

weigh heavily with (one)

To be an important or influential factor for one, especially when faced with making a decision. This is an issue that weighs heavily with my constituents, so I am making it a priority during my time in Congress. Being close enough to visit my family during breaks weighed heavily with me when I was choosing which college to attend.
See also: heavily, weigh
Farlex Dictionary of Idioms.

weigh with

v.
To be of importance to someone when making a decision: The issue of taxes will weigh heavily with the voters.
See also: weigh
The American Heritage® Dictionary of Phrasal Verbs.
See also:
  • weigh heavily with (one)
  • high ups
  • higher ups
  • higher-ups
  • high-ups
  • high muckety muck
  • high mucky-muck
  • muckety
  • high muck-a-muck
  • muckamuck