I can't think of a "cut/slice" idiom to describe a person who does not have clout/influence, but in America we do say "That won't cut it" to describe something that is not good enough to solve a problem or meet a standard. Example: "That rope you brought won't cut it. It's not strong enough enough." Another, slangier expression for no clout is "no juice." Example: "Your mentor doesn't even have the juice to get a promotion for you. You need a mentor with more juice."

I can think of a couple of examples using the word 'cut', both of them usually used negatively. 'It doesn't cut the mustard' means 'it doesn't do the job' or 'it's inadequate' (like 'that won't cut it' in John Brooks' answer above), and 'it cuts no ice with me' means 'it doesn't impress me or have any influence on me'. Nobody seems to know for certain what the origin of either phrase was.

'He's got no clout' in the US would suggest 'he's got no authority/weight' which doesn't seem to match the meaning of the Spanish expression you mentioned. Not sure exactly what you are looking for but perhaps something like 'a dull blade' or 'not the sharpest tool in the woodshed'?

Not off the top of my head using a cutting word. "He's got little influence"... would be a very polite and formal way of saying "he's got no clout" if that's of interest.