I think that the best word in this case is "влияние". For example, "Это очень влиятельный епископ".

The word власть is used when you are talking about official capacity or ability to have control/power over someone or something, usually in political or administrative context. Eg: у американского президента много власти / в руках президента много власти. In other cases, you can say: ''у этого менеджера много возможностей'' (возможности here means abilities) or ''это очень влиятельный менеджер''. The adjectives ''влиятельный'' and ''influential'' have a slightly different connotation. ''Влиятельный'' refers only to the ability to have an influence in decision making, to play an important role in the management etc.; it is not used in contexts like for example: ''it was a very influential books'', you can't say ''влиятельная книга'', but ''книга, которая повлияла на многих''. The word мощь is quite rare, you can find it in phrases like ''военная мощь'' or maybe sometimes ''экономическая мощь''. Властный means commanding, having the tone of authority (eg властный голос, властная манера говорить) Авторитетный -- authoritative, highly reliable, respectful, having a lot of knowledge or experience in a particular subject.

Властный/сильный/авторитетный/влиятельный

I'm not sure what exactly you mean, but there is one more word - "могущественный".