Is it genitive because it is an animate noun, or because wspierać requires the genitive?
That's actually Accusative, (Genitive would be 'tego mężczyzny') and is required by 'wspierać'.
it is an male animated noun for persons, but with a female ending. the article is changed as if it was a normal animated male noun for persons ( like : ten syn, tego syna ( for gen. and acc. ) but the word meyczyzna acts like a female word therefore it has the ending ę in accusativ.
Actually that'd rather be "Popieram". This sounds more like this is someone I know personally and I support him... financially, emotionally, by helping him in some other way?
Although if you were really working with this politician, in his campaign, "wspieram" could probably work. Or if you donated to his campaign. But for a normal voter, who can only vote, I wouldn't use it.
Does wspieram mean pysical support (leaning an arm) or financial (sending money)?