There's nothing to say you can't do both. Some banks don't deal with brokers at all (NIB for example, although their rates aren't too hot at the moment). I'd get a list from the broker of the banks s/he is going to approach and then go to the other ones yourself. It's a bit more work to do it yourself but it gives you a better insight into the workings of the whole thing.
It's impossible to know what, if any, influence a broker will have on the banks. Obviously, it will come down to your income and what you can repay - I don't think that a "reference" from a broker will sway things much either way. Banks don't have to know you personally to consider your application - over the years, I've gotten quotes from the three banks that I use and haven't once met someone in the bank that I'd met before.
The thing with a broker is that you have to be sure of the banks they are submitting your application to. If you trust the broker, then that won't be a problem. You will always have the gap between the banks that deal with a broker and those who don't though. I'd take a look at the interest rates and see which ones suit you the best and make sure the broker approaches those institutions and, if they don't/can't, then do it yourself. Using a broker takes a lot of the hassle out of things, but only works if the broker gets you the mortgage you want.