It is true that most of Circassia was never brought under direct Ottoman rule. However, you may have been misled by maps depicting the eastern coast of the Black Sea as being outside of their control into thinking that they had less engagement with the region than was actually the case. Ottoman influence over Circassia began in the immediate aftermath of the conquest of Constantinople in 1453, during the reign of Sultan Mehmed II (r. 1451-81), also known as Mehmed the Conqueror. One of this ruler's goals was to establish Ottoman hegemony over the Black Sea, with the empire's primary adversary at sea being the Genoese, a merchant republic that held numerous colonies across the region. The most significant of these was the Crimean port of Kaffa (conquered by the Ottomans in 1475), but their holdings also included ports in Abkhazia and Circassia. Given that the Ottomans now controlled the only maritime entrance to the Black Sea, these were easily conquered, and became the basis for Ottoman control over the region. Coastal Abkhazia and Circassia did not constitute a separate province, but were annexed first to the province of Rum (i.e. Sivas), and later to the province of Trabzon.

However, having control over a handful of ports is not the same as conquering the entire region. Ottoman rule was basically limited to these fortified positions. The political and economic structure of Circassia would also have made direct control difficult and unattractive. Circassia had a mixed economy of agriculture and pastoral nomadism, and the region was not densely populated. Politically, it was divided into a number of competing princedoms and clans. Lords with the hereditary title pshi, usually translated into Ottoman Turkish as mirza, constituted a form of nobility. Ruling over Circassia would require incorporating these lords into some kind of relationship with the state, which the Ottomans attempted to do by providing them with gifts and stipends (a continuation of Genoese practice). However the extreme decentralization of power in the region made more intensive integration difficult. The lords were constantly in conflict with one another, with no single individual able to impose his authority over all the others. As a result, the relations between the Ottomans and the various Circassian lords were continually in flux, with no long-term, stable arrangement emerging that could have tied them more closely to the Ottoman state.

The other factor limiting Ottoman sovereignty was the political influence of the Crimean Khans. As heirs to the legacy of the Golden Horde, the Crimean Giray dynasty claimed authority over Circassia, and sought with varying degrees of success to subordinate the various Circassian lords to them personally, rather than to the Ottomans. The Crimean Khanate is itself often thought of as having been a "vassal" of the Ottomans, but the degree to which this was an accurate description varied. Especially in earlier periods, prior to the eighteenth-century emergence of Russia as a dominant force in Eastern Europe, the Crimean rulers could act with a significant degree of independence. The Crimean rulers established themselves as mediators between the Ottomans and the Circassian lords, mobilizing the latter for war both in Hungary and against Iran. Some of these warriors decided to stay in the Ottoman Empire in the aftermath of these campaigns, learning Turkish and becoming part of the Ottoman standing army. Ottoman and Crimean influence over the region led over the course of the early modern period to its gradual Islamization. In the 15th century most Circassians had been at lest nominally Christian, but by the 19th century they were almost entirely Muslim.

Because of the difficulties that would arise from attempting to impose direct rule, Ottoman engagement with the region primarily took the form of trade, through its export of cereals, salted fish, caviar, honey, beeswax, lumber, wool, animal hides, and furs. Most significant of all, however, were slaves. Circassia and Abkhazia's prominent place as a source for Ottoman slaves stemmed from two factors already mentioned: the constant conflict between feuding clans, and the region's incomplete Islamization. Clan warfare produced a constant stream of captives, and lords could profit from selling enslaved individuals to the Ottomans, much as they had previously sold them to the Genoese. Ottoman-controlled towns on the coast derived much of their prominence as centers of this slave trade. At some point the Ottoman palace itself began purchasing Circassian and Abkhazian slaves to function as servants. Historians usually date this phenomenon to the late sixteenth century and see it as an imitation of Mamluk practice, adopted some decades after the conquest of Syria and Egypt in 1516-7. My own research points to a somewhat earlier origin, because Circassians and Abkhazians were already common in certain segments of the early sixteenth-century Ottoman standing army, which they would have joined after completing their years of palace service. While most palace servants eventually joined the standing army, some were promoted to more prestigious functions within the palace, becoming closer to the person of the sultan. The closest of all - those who had the privilege of overseeing his privy chamber, carrying his sword, serving as his master of the hunt, etc. - could hope that when they left the palace they would be promoted to higher ranks of the Ottoman elite, becoming provincial governors. In this manner, several Circassians and Abkhazians who had originally been purchased as slaves were able to work their way to the top of the political hierarchy. Several grand viziers over the course of the seventeenth century were of this origin. As the sultan's subjects could not normally legally be enslaved, conquering Circassia would have brought an end to this region's ability to supply slaves to the Ottoman elite.