I think a simple way to improve your time would be if you could run the whole track. I myself am not running as regular as you, so I cannot tell you how fast you should see improvements, but after I horribly failed to run a consecutive 5km when I was running for the first time, I tried to step down a bit.
The program I would advise you to look into is called the 3:2:1 program covered by @JohnP in an answer to one of my questions.
I would also recommend a program created by a cross country coach named Barry Pollack, dubbed the 3:2:1 program. In this, you have 6 runs per week, 3 short, 2 medium and one long. Your medium run is double your short runs, and your long run is 3x your short run.
I never really did the 3:2:1 program myself as I wasn't running every day, but I applied some of the aspects to my training. Depending on your long term goal after you reached the goal stated in your question you might want to modify how you train, too.
I would, in your case, try to do a shorter but consecutive run on one day and the long run with walking on the other day. You should also put a bit of variation into the choice of your track, simply to prevent falling into patterns "This is the point where I always start to walk; I need to walk now".
Have a look at the Couch 2 5k Program, you might find some other ways to improve yourself in it, too. Although you probably don't need to start in week one or two, be careful and don't overestimate yourself. It is sometimes better to step down a bit to be able to change the approach to a workout.