How important is it that your hero have "feet of clay"?
For me it's not just important, it's crucial if I want my heroes to be believable. Some of the best advice I ever got about writing comes from Chuck Dixon, and it has served me well throughout my career: Give your villain a likeable trait, and give your hero a dislikeable trait. Those aren't his exact words, but that's the gist of it.
Give your hero feet of clay. Make them struggle with something, and for my money, why not a moral issue?
Case in point, Rick Ruby from THE RUBY FILES is a womanizer and a bit of a drunk, but he gets the job done and has a moral code that defines his world. Armless O'Neil (from my upcoming book from Pulp Obscura) is a cynical and jaded bastard who tries to overcome his lack of faith in people to ultimately do the right thing. And in my SHOW ME A HERO collection, the Grandstander is a flamboyant and arrogant jackass who's only super power is that he knows the exact moment of his death and has a sort of carte blanche with his actions until that day happens, and he abuses his "power" to impress people. But not without ultimately doing the right thing.
Traditional pulp, I'll admit, is probably one of the most difficult genres for me to write, simply because it's well-known for its clear cut delineations of good guys and bad guys, whereas I prefer to write my characters along a continuum rather than a clear line in the sand. My bad guys can be a little good, and my good guys have to be a little bad, or they're just not that interesting to me as a reader or as a writer.