Being able to call down miracles regularly and often is a trump card for religious factions in strife. A good example comes from the (Christian) Bible. See the story of Elijah and the Priests of Baal in the Bible.
If these miracles are too common, people will get used to them, and may not take them as miracles anymore. This is especially true if other religions can perform those miracles, too, as seen in the biblical account of Moses calling down the Plagues of Egypt.
Scientists would take the miracles as fact. Some could say "those are not miracles, it's science." Others could investigate these miracles and take a scientific approach to them, figuring out their laws and mechanics.
It would certainly give science a goal; to figure out why those priests can call down those miracles! This is, of course, assuming there is a division between science and religion in that culture.
If priests can call down miracles regularly or on-demand, and those miracles are taken as divine recognition of those priests, then that religion must be correct. No errant ideas or philosophies which counter that religion should exist, unless those philosophies can produce those miracles as well (via priests who subscribe to them). It would be a litmus test to see if those philosophies were correct or not.
Miracles on-demand would be demanded. Whatever these miracles can affect would be very different. This could lead to a market for purchasing miracles. The price would be determined by the availability of these miracles. People would come to rely on miracles the same way we have come to rely on modern technology. In some cases, the demand may result in people forcing priests to perform that miracle or face consequences.
If the miracle is common enough, why develop the technology to do it? Sure, you'd not be dependent upon a priest, but if those priests are everywhere, it may not be worth the resources to create that technology. That's the opportunity cost of regularly occurring miracles.