Good day Duncan. To improve your soil's water holding ability you will need to add some sort of organic matter to the native soil that you already have. Peat is an alright choice, but it has a low nutritional value, where as compost will add organic matter and nutrition.
Suggested action:
Measure the square footage of the lawn area in question.
Next figure out how much organic material you will need. To do this you will need to know that a cubic yard will cover 108 square feet a 3" deep (10 foot X 10 foot square). You will want to add at least an inch with 3 to 6 inches being the best ammount to have if possible.
Survey the area and see if the amount of new soil will alter the grade too much around your house, patio, trees, driveway ect.
Rototill the lawn area first, then again after the new soil/compost is layed. Rake out the lawn area after the tilling then roll the lawn with a lawn roller. This is an important step that many folks leave out. Re-seed the area and cover with hay to keep the birds out of it untill it sprouts. The hay will multch up when you mow.
we use compost from the mulch place, they bring it in dump truck..it has some sand in it too. last year i worke 20 yards of it in my garden too.. you can till it in with a rear tine tiller , if it large area ,use tractor and disc... or you could just get topsoil
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