East Yorkshire sounds Saharan! I do appreciate your point and recent springs have been much drier than summers.
I may not be the best person to answer as we have a little more rain than you describe, for example in March we had 1.2" (30mm) and in April so far, until 11th, we have had 3/4" (19mm), just enough to maintain moisture although some plants like leeks and spring cabbage could do with rain.
On the other hand i do find that animal manure has helped, and retains more moisture than the fluffier green waste and mushroom composts. I gather you do nit dig, which is the main way of maximising early moisture. I avoid bulky mulches like straw because of potential for slugs and their impracticality in terms of sowing and setting out small plants.
Re irrigation, I have never practised it so cannot help there. I do all watering with a hose, even in my tunnels, about half an hour a week in each one at the moment. All plants are watered in and then left to send their roots down, except in great heat when a second or third small watering is given, always by hand to the rootball, not over all the soil, so water is economised massively.