There certainly were! First, let's define what we mean when we speak of mercenaries in the Ottoman military context. Historians like to speak of the existence of a "classical" or "traditional" Ottoman army that came into existence over the course of the fourteenth and fifteenth centuries, and reached its peak in the early sixteenth. This army was defined by the conjunction of two distinct forms of military organization: the palace troops (also referred to as the sultan's "household troops", in Ottoman Turkish kapukulu) and the timariots. The palace troops were a standing army of salaried soldiery divided into multiple branches consisting of infantry (the Janissaries), cavalry (the so-called "Six Regiments"), artillerymen, and a variety of auxiliary branches such as armorers and wagon-drivers. The timariots were initially much more numerous, and consisted of a body of cavalrymen paid through grants of agricultural revenue (called timars) from landed estates spread out across the empire. These were the two central pillars of the Ottoman army in the fifteenth and sixteenth centuries, although other auxiliary forces were also extant. So in early modern Ottoman history, when we speak of mercenaries, what we mean are soldiers who operated outside of this system, who were paid in cash like the kapukulu/palace soldiery but who weren't officially part of that organization, and therefore lacked their social prestige and political/economic privileges, as well as their security of tenure: unlike kapukulu, mercenaries would only be paid sporadically, hired for a campaign and then dismissed when the fighting was done.

The terms used by the Ottomans (and thus by modern scholars) to refer to such troops were levend, sekban, and sarıca. Each of these words could have more specific meaning in particular contexts and could sometimes refer to people other than mercenaries, but for our purposes we can treat them as synonymous in meaning "non-kapukulu soldiers who fought for whomever would pay them in exchange for money." These groups had their origin in the mid-sixteenth century: we first find mercenaries (then called yevmlü, "daily-wagers") being hired in large numbers during the civil wars between the sons of Süleyman the Magnificent, but the use of mercenaries began to seriously take off starting in the 1570s. We start to find Ottoman armies employing mercenaries on a larger scale, with units being hired either directly by the state or by the empire's high-ranking provincial governors to serve in their private armies. There are a number of reasons that have been proposed to explain the sudden emergence of this phenomenon:

1. The growth in population in Anatolia over the course of the sixteenth century, which led to greater landlessness and bachelorhood, pushing young men into military professions or outright banditry (the poverty of the countryside being further worsened near the end of the sixteenth century due to the effects of the Little Ice Age)
2. The spread of handheld firearms, which had previously been rare but now became easily accessible.
3. The expansion of the money economy, increased availability of ready cash being a prerequisite for the use of mercenaries on a larger scale, both by the Ottoman state and by non-state actors.
4. The demand for manpower created by warfare against Habsburg Austria and Safavid Iran, particularly after the Ottomans became engaged in fighting both simultaneously around the turn of the seventeenth century.

In explaining this phenomenon, there is a general division among historians between those who emphasize "push-factors" such as the poverty and overpopulation of the Anatolian countryside, and those who emphasize "pull-factors" such as the desire of the Ottoman state to acquire more manpower for its wars. Where historians agree is in seeing a general relationship between the emergence of large-scale banditry in Anatolia and the increased role of mercenaries in the Ottoman armies. The so-called "Celali Rebellions" of the turn of the century drew heavily upon mercenary manpower: an unemployed mercenary would have few economic options to sustain himself other than to engage in pillaging the countryside either in small groups or under the banner of a local warlord, and likewise bandits already armed with firearms made good cannon fodder for the frontiers with Iran and Austria, thus a mercenary/bandit could switch between these two roles quite easily.

For the mercenaries themselves, the ultimate dream would be to one day be accepted into the ranks of the kapukulu. This would mean gaining social respectability, a steady salary, exemption from taxation, and the expectation of a retirement pension in case of injury or old age. There were a number of ways to achieve this: showing exceptional valor on campaign, signing up as front-line assault troops called serdengeçdi ("he's out of his mind") and surviving the subsequent battle, or earning the favor of influential people. Most could not do so, however, and would be left to fend for themselves when not on campaign.

We don't know much about how individual mercenaries were hired because they were organized into units (sing. bayrak, "banner"), each with a captain who would be in charge of managing his followers, and who would receive and distribute the entire unit's pay. These units would range from 50-100 men ideally: I'm in possession of a logbook of expenses from Murad IV's 1638 Baghdad Campaign that lists sekban units ranging from a low of 31 men to a high of 81, and sarıca units ranging from a low of 62 to a high of 115 (here the sekban/sarıca distinction likely referred to cavalry vs. infantry units). These units had a high turnover rate in employment: a surviving record of the income and expenses of the governor of Diyarbekir in 1671 shows that any given mercenary unit could only expect to remain attached to their employer for a few months at a time: they would be hired according to need and dropped as soon as possible. We know nothing about the turnover rate of manpower within these units, however.

Shortly after their appearance, mercenaries became a hot-button political issue in the empire. Many kapukulu recruited through the traditional means (such as the devşirme, the conscription of Christian boys from the Balkans into service in the Ottoman imperial household), or descended from those recruits, saw the attempts of mercenaries to join their organization as an unwelcome intrusion. Worse, they saw the mercenaries as a political threat, because the military power they could offer the sultans might render the kapukulu obsolete. One sultan, Osman II, was murdered in a kapukulu rebellion in 1622 because of the rumor that he intended to replace them with a mercenary army raised from Anatolia. This led in turn to the uprising of the governor of Erzurum, Abaza Mehmed Pasha, who used his mercenary army to avenge the sultan by hunting down and executing all the Janissaries he could find.

So not only were there mercenaries in the Ottoman Empire, but their presence created enormous tension and even led to the murder of a sultan. We can see in them frustrated young men looking to make a living for themselves in the economic crisis years of the seventeenth century, hoping to find a way to be accepted into the ranks of the Ottoman elite as kapukulu, and for the majority for whom that door of opportunity never opened, bandits ready to prey on the countryside until the time came for the next employment.

For those who are interested in this topic, the definitive place to start is Halil İnalcık's article "Military and fiscal transformation in the Ottoman Empire, 1600–1700." Archivum Ottomanicum 6 (1980): 283–337. This is a seminal article in Ottoman history, drawing attention to the emergence of the levend/sekban/sarıca phenomenon as critical for understanding the changing nature of the empire, its military, and its administration in the seventeenth century. Although outdated in some respects, it remains an indispensable classic.