It varies by period, and to an extent what you mean by influence. The initial Caliphs (the Rashidun, the early Ummayads) seem to have had quite a lot of leeway to promulgate religious as well as political law. This perceived absolutism by the time of the later Ummayads prompted a reaction from both the burgeoning Ulema, the religious scholarly class that was, in effect, codifying the Sharia, as well has the Shia who held that such religious authority could only be vested in a genuine Imam, one of the direct descendants of Ali, in a doctrine that came to be known as "Imama."

The result, in a sense, was the Abbasid revolution. While the early Abbasids would likewise attempt to guide religious law and direct the Ulema the tension in that system would eventually come to the fore in the Mihna (inquisition) of the Caliph Mamun, particularly over his attempt to enforce his own theological position of the createdness of the Quran among the scholars of the empire. While Mamun's successors briefly carried on the Mihna it was eventually stopped having had little influence. There would be no real additional attempts on the part of the Caliphs to shape religious law and their powers were restricted to more political legislation.

edit: It's also noteworthy that this restriction was effectively "retconned" to apply to the Rashidun as well. Despite us not having any evidence that they were doing anything other than introducing new religious law, it has been held by the Ulema ever since that if they were doing so it was in their capacity as Sahaba, companions of the prophet Muhammad who were simply emulating the prophetic example, just as later scholars were attempting to do. Ummayad rule, in contrast, was denounced as mere "Mulk", secular kingship.

A further change came again when, with the rise of largely Turkic dynasties who established themselves as Sultans (e.g. the Buyids, the Seljuks, etc), the Caliphs were effectively deprived of political power as well. What was left was essentially a religious figurehead position. They were the head of a community of believers over which they had no power to shape that belief.

Political and religious authority would eventually be re-joined in the Ottoman Caliphate, but Ottoman authority was derived from their position as Sultans. I'm not aware of any attempt of theirs to promulgate religious law. When their own legislation impacted on religious law it was done as a form of civil law, not as an attempt to alter the Sharia.

Source wise:

God's Caliph by Patricia Crone is an excellent monograph on the evolution of the powers and concept of the Caliphate, particularly in the early period.

For the position of the Ulema vis a vis the Caliph in wider Islamic society, and how these religious scholars came to occupy a position where they established independent legislative (or at least judicial) control I would have a look at Marshall Hodgson's The Venture of Islam, or from a more technical perspective something like Joseph Schact's Introduction to Islamic Law.

For Ottoman law, I'm not sure I can point to a particularly good individual source, but the relevant topic is the concept of "Qanun" law.