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Should I try and do a couple of short, fast runs as well?
Yes. Intervals and longer slower [than interval pace] but still fast[er than race pace] paced runs will help with your speed. I find one interval session and one tempo session a week to be beneficial, and you'll find a lot of training plans with similar. Then add in your long run to focus on building the distance, run at slower than target pace and you'll be set!
Training for my second half marathon coming up in October. I want to get a more competitive time than my previous 1:44 but not sure the best way to go about it. Current training is roughly one run to work a week (11-12km) and one after work a week (7-8km). Should I try and do a couple of short, fast runs as well? Or is it just about building up and getting used to distance?