their beliefs and worldly views

I'm not sure that these would be possible to generalize into an actual set of beliefs and views. The etymology of "Sufi" is disputed, but perhaps the most plausible origin of the word is as a reference to the rough woolen garb worn by Islamic ascetics, comparable to the "sackcloth" or "hair shirts" worn in some Christian traditions.

While asceticism can be a component of Sufism it actually covers an enormous range of ascetic and mystical practices. If I had to give a one-sentence of what Sufism is, I would probably describe it as a catch-all term for Islamic mysticism, most typically though not entirely exclusively Islamic Sunni mysticism.

As a result, the short answer to your question is "Yes" Sufis and Sufi orders have been incredibly influential in the politics of Muslim states.

There are different ways to look at this, so many in fact that I'm not sure they can all be summarized into a single response, but I'll give a few examples.

• Though there is some dispute about this, Sufis seem to have played an important missionary role in converting the Turkic tribes on the peripheral territories of Islam prior to the Turkic conquests of central Islamdom (e.g. the Seljuks). The eventual Turkic rulers themselves were generally quite orthodox, which is one reason why there's debate as to how important the Sufi role was.
• By the late Medieval and early modern periods in the Islamic world, the institutionalization of Sufi orders, tariqas, and the building of Sufi lodges, khanqas, was a critical element of civil society and order across the Islamic world. Off the top of my head I can't recall the specific figure but some massive percentage of adult males in the Ottoman empire would have been associated with a tariqa. Each profession or trade would have had a particular association with a given tariqa. This included the Ottoman government itself, with the Janissary corps, for instance, being closely associated with the Bektashi Sufi order.
• Safavid Iran had its origins in the Safaviyya Sufi tariqa.
• Various revivalist movements in Islam can be characterized as Sufi or perhaps as "Sufistic", although they might reject the label themselves. The Afghan Taliban, for instance, is closely associated with the Deobandi movement, which might be called Sufi.

Again, these are just a few examples. But I think the overall answer to your question would have to be "yes."

Perhaps more interesting is actually the decline of Sufism over the past 200 years or so, which happened for a number of reasons including a general end to state sponsorship or outright state hostility to Sufi practices (e.g. in Turkey, where many such movements were banned outright. The famous "Whirling Dervishes" of the Mevlevi tariqa of Konya, if I recall correctly, escaped censure by basically rebranding themselves as historical/cultural dance association.)

The general decline in Sufism is also associated with the emergence of Salafism and certain kinds of revivalist Islam and with the "Saudization" of Islam over the past century in general, which is hostile to Sufi practices.