The phrase, "zone(s) of influence" is a likely forerunner to the contemporary sounding phrase, "sphere(s) of influence".
When The Ottoman Empire collapsed at the end of the First World War, Western Asia, was essentially, "up for grabs"-(geopolitically speaking or so the West European countries thought). Of all the areas of the Ottoman Empire, it was Western Asia-(what was called, "Asia Minor" or "Anatolia" during Pre-Ottoman times) which had the largest percentage of ethnic Turkish Muslims-(as well as other ethno-religious minorities, such as Kurdish Muslims, as well as Greek, Armenian & Assyrian Christians) residing in this region. The WW I victors, namely the British and the French, had established earlier "zones of influence" with the neighboring country of Greece, as well as having a strong partnership with Italy-(also a British and French ally during the First World War).
By establishing "zones of influence" within West Asia-(or the beginning point of the Middle East), the British and French could either directly occupy the Middle East by balancing their colonial power or (as the British would do during the reign of Egypt's King Farouk), would establish puppet regimes who were friendly to British and French colonial interests-(The "Our Man in x country" policy).
In the case of Western Asia, it was-(and is still), a land and region rich in waterways, as well as land resources, thus, an ideal center for trade/ commerce. For the British, the West Asia "zone of influence" would expand their colonial state and in turn their "influence" throughout the massive continent of Asia, while also being in close proximity to the beginnings of the African continent-(through Egypt). Instead of a defeated and politically beleaguered Turkish state continuing its own centuries old "zones of influence" throughout wider Western Asia, the French and particularly, the British, would emerge as the new colonial puppeteers.
By also allying with the Italians and the Greeks, the French and British could essentially, Europeanize the beginning point of Asia in commercial, military, diplomatic and even cultural terms. The 1920 Treaty of Sevres would help to pioneer the Modern "zone of influence" or "sphere of influence" concept, though it was, historically speaking, a short-lived policy that resulted in abject failure. The result, did not produce a stable and governable demarcating of British, French, Italian and Greek zones of influence, but, instead, unwittingly galvanized a rabid and ruthlessly determined Post-Ottoman Turkish Nationalist movement which would conquer every European zone of influence within Turkish West Asia, thereby establishing the Modern nation-state of Turkey in 1922.