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.
To be of importance to someone when making a decision: The issue of taxes will weigh heavily with the voters.
See also: weigh