Yes you can and should water your lawn after you mow it.
Just be sure to wait to water the lawn early in the morning or late in the evening when the sun is not out and shining too hot.
Watering the lawn in the sun when it's hot out can cause the lawn and grass to get burnt and also cause the water on the lawn to evaporate before the lawn has a chance to drink it up.
It's okay to run the sprinkler in the hot summer on the lawn especially when kids are playing in the sprinkler water.
But if your main goal is to water the lawn and get a nice healthy looking lawn then it's always best to water the lawn in the early morning hours or late evening as the sun is going down and when it's cooler out.
I always turn the sprinkler on my lawn after I mow it and it helps the water get into the lawn and soil much easier as well.