ethelruns,
Generally you would focus on increasing your mileage first. With three runs a week I would for now work on increasing the length of your runs, gently to 5-6 miles and a long run of 10-14 miles. You can do this over a few months time, you have some time until your autumn half marathon.
Given you are limited to three days / week of running training and want to do the speedier work I would then probably replace one of the shorter runs with tempo runs predominantly. In the half marathon the speed you will be using is below your lactate threshold, so I would, as a beginner, definitely focus more on the tempo runs. Maybe have 2 weeks in which you do tempos, then 1 week in which you do intervals.
I would advise against removing more easy runs out of your schedule in favour of the faster stuff, easy running should be the cornerstone of your program, especially when you are still working towards being able to cover the distance.
Hope this helps.
Kind regards,
Best Running Tips