Too many variables involved to give you a very accurate answer.......depends on the type of mulch and how thick it has been applied and whether or not it has been allowed to dry out; depends on how the sprinkler system is positioned and programmed; and finally it depends a lot on the soil texture and how moisture retentive it is.

Generally speaking, a low emission water delivery system - like a drip system or a soaker hose - is a far more efficient form of irrigation than pop up sprinklers, as it delivers water directly to the root zone where it is best utilized by the plants. Pop ups tend to waste a good deal of water through evaporation (estimates run as high as 75%, depending on the system and its programming) and can lead to the spread of pathogens by the splashing of spores, etc. or water remaining on foliage for extended periods.

But many folks do irrigate successfully with an automated pop up system. Just make sure the water remains on for a long enough period of time (once or twice a week for 20-30 minutes at least, rather than daily for 5-10 minutes) and that the mulch does not dry out excessively and become hydrophobic. You may need to do some testing to determine how deeply the water penetrates the soil profile to figure out the proper pop up programming schedule. You want it to penetrate the full depth of the root ball if possible or somewhere around 8-10 inches with each watering.