influence vs bear upon

influence is a noun but bear upon is not a noun.

Word	Noun	Adjective	Verb	Adverb
influence	Yes	No	Yes	No
bear upon	No	No	Yes	No

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

Other hypernyms of influence include affect, bear on, impact, touch, touch on.

influence (noun)	bear upon (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

influence (verb)	bear upon (verb)
shape or influence; give direction to	have an effect upon
induce into action by using one's charm
have and exert influence or effect