I started from a similar position to you about two years ago but today did a PB of 51m in a 10k race. I have never done any specific speed work but just plodded along and as time has gone on and I've got fitter you get faster.
My philosophy is always to put a similar ammount of effort in, you don't want to end a run feeling like you could have run it faster, you want to end it out of breath, etc. As your heart and legs improve to put the same effort in, you have to run faster.
What races are you entering anyway?
At our level of ability we're never going to win any races, so really you're just racing yourself and your goals. My general aim is to finish a marathon at least half as fast as the winner or finish a race halfway up the finishers.
With practise your time will come down, your baseline is probably the best to improve from if you stick at it you'll be able to knock 30m off that HM time.
R