carry weight vs influence

carry weight is not a noun while influence is a noun.

Word	Noun	Adjective	Verb	Adverb
carry weight	No	No	Yes	No
influence	Yes	No	Yes	No

As verbs, influence is a hypernym of carry weight; that is, influence is a word with a broader meaning than carry weight:

Other hypernyms of carry weight include determine, mold, mould, regulate, shape.

carry weight (noun)	influence (noun)
	causing something without any direct or apparent effort
	a power to affect persons or events especially power based on prestige etc
	a cognitive factor that tends to have an effect on what you do
	one having power to influence another
	the effect of one thing (or person) on another

carry weight (verb)	influence (verb)
have influence to a specified degree	shape or influence; give direction to
	induce into action by using one's charm
	have and exert influence or effect