carry an amount of weight

carry (an amount of) weight

To wield importance or influence. Due to your status, your remarks carry a lot of weight, so I encourage you to speak out on this issue—people might finally start taking it seriously. Without research to support these claims, I'm afraid they don't carry much weight.
See also: amount, carry, weight
Farlex Dictionary of Idioms.

carry (an amount of) weight

tv. to have a degree of influence. Tom carries a lot of weight with the mayor. Ask him.
See also: amount, carry, of, weight
McGraw-Hill's Dictionary of American Slang and Colloquial Expressions
See also:
  • carry (an amount of) weight
  • carry weight
  • carry weight with (someone or something)
  • throw (one's) weight behind (someone or something)
  • throw your weight behind someone
  • throw your weight behind something
  • a pep talk
  • shotty
  • shotty back
  • in the country of the blind, the one-eyed man is king
References in periodicals archive
Working alongside facilitators from Babcock Marine and the Engineering Sciences Department at the University, students were set to work on the main hands-on design-and-make project to build their own ship against set criteria and strict specifications such as the ability to carry an amount of weight and to travel at a certain speed.