Keep on mowing....
Lawn care in June is mostly about the mowing schedule, stepped up to twice a week if the conditions are promoting growth. You can still treat the established lawn containing weeds and moss with a lawn fertiliser, weed and moss killer such as Rolawn and remove by hand or spot treat coarse grasses and weeds.
Aerating or spiking the lawn will help get water and air to the roots in dry periods keeping in mind the summer droughts we have been experiencing.
If you are thinking about creating a new lawn you can prepare the area now but the lawn is best started in August to September.