I'll try to answer your question, sorry if it doesn't satisfy you. Please be noted though that we don't actually have much records of New World crops in the Ottoman Empire. Most of we have are inventory and purchase lists, which don't tell much about how were they used. In any case, i'll address the crops one by one

Pumpkin

The American pumpkin first appeared in palace kitchen accounts with the name ‘Egyptian gourd’ in 1555. Pumpkins became so popular that in 1594 more than a thousand were purchased at Manisa near the Aegean coast for the kitchen of the future Mehmed III. At first, pumpkins were consumed by elites, as pilav, soup, stew, or made into jams. For example, a banquet in mid 17th century Topkapi Palace listed a pilav with pumpkin and honey. As time goes on, pumpkins made it into consumption for common people, though it's unknown how widespread it is. Pumpkins frequently appeared in imaret (soup kitchens) menu to be served to the poor and people who deserved it. The imaret of Fatih complex served aşraf (descendants of Muhammad), with sheeps’ trotters (paça) served for breakfast as a great delicacy, as well as a dish made of pumpkin, honey, jam, cinnamon, and cloves. The same imaret had records of serving at least 60 kilos of pumpkins everyday

Tomato

We don't know the exact date of its appearance, but the earliest reference to tomatoes (kavata) in the imperial palace was in 1694, where some 13,350 pieces (aded) of them -which was something like a ton- were stored. These first tomatoes were green tomatoes. In 1704, Ottoman palace authorities distributed 450 kavata on a daily basis. While 150 fruits went to the kuşhane, a private kitchen at the Harem, the rest was given to distinguished individuals including the chief black eunuch of the sultan (darussade ağası), the treasurer (hazinedar), the female chamberlain (kethüda kadın) and the chief consort of the sultan (aşkadın), aside from a number of institutions. These recipients also enjoyed allocations of kavata leaves. Starting in the 1750s, a new vegetable, called frenk badincanı or “Frankish eggplant” entered the markets of the Ottoman capital. According to 19th-century dictionaries, this vegetable was a tomato. As the term kavata still occurred in Ottoman archival documents of the late 1700s and early 1800s, frenk badincanı should have been a different type, namely the red tomato. By the late 1700s, both green (kavata) and red tomato (frenk badincanı) were present in Istanbul markets. As time goes, tomatoes becomes a staple food and the term frenk badincanı were replaced by the domates. Kavata were generally used to make pickles and give sour taste to dishes, while domates were popular in meat stews (yahni), in vegetable dishes, stuffed pepper, salad and pilaf, and made into paste. Both kavata and domates appeared frequently in 19th century Ottoman cookbooks. Melceü’t-Tabbahin, the first Ottoman cookbook published in 1844 included recipes for a tomato stew (domatesli yahni), stuffed tomatoes (domates dolmasz), tomato pilav (domates pilavz), and a tomato salad (domates salatasz). A 1892 cookbook by Aşye Fahriye included an impressive amount of domates and kavata recipes: kavata stew with meat in small casseroles (kavatanm kuşhane musakkasz), kavatas stuffed with meat (etli kavata dolmasz), and pickled kavatas (kavata turşusu), tomato stew (domatesli yahni), tomatoes stuffed with mussels (midyeli domates dolmasz), a large-tray tomato stew with meat (domatesin tepsi musakkasz), garnished tomatoes (domates garnitürü), a cold stew of mussels in olive oil with tomatoes (domatesli midye pilakisi), a cold stew of tomatoes in olive oil (domates pilakisi), pasta with tomatoes (domatesli makarna), tomato pilav (domatesli pilav), and tomato salad (domates salatasz).

French Beans

French beans originated from the New World, cultivated and eaten in Middle and South America both before and after the arrival of the Spaniards. The cultivation of French beans in Istanbul and surroundings began in the first half of the 18th century, as the earliest Ottoman record goes back to 1739. This document stated that gardeners brought these vegetables to the customs house known as Gümrükönü and sold them to greengrocers and other salespeople in the cellar of the bamyacıbaşı. This meant that French beans are not only popular, but also steadily becoming a staple food for some people. After about 1750, French beans also appeared in the lists of administratively determined prices (narh), another testimony to their growing popularity. Narh registers of the 1800s referred to various kinds of beans, including string beans (çalı fasulyesi) and black beans (kara fasulye). Towards the end of the century, another type of bean, called Ayişe kadın, appeared in the records as well; and in the very early 1900s, among the foodstuffs purchased for an unknown palace in Istanbul, three types of beans were on record: dried haricot beans (kuru fasulye), string beans (çalı fasulyesi) and ordinary beans (adi fasulye). After about 1850, the demand for this vegetable increased so dramatically that the surroundings of Istanbul, the traditional production center could not meet the demand. Therefore, Anatolian regions, including Trabzon and Samsun began to provide beans to the capital. As for the Ottoman Balkans, Bulgaria, especially Varna, produced beans for the Istanbul market too. Consumption increased further because beans became a staple food for the numerous soldiers of the Ottoman Empire.

Chili Peppers

American pepper (chilli pepper) was however introduced to Europe by the Spanish physician Diego Alvarrez Chanca, who wrote an article about the medical benefits of this plant. The first Istanbul records concerning this plant date back to the second half of the 18th century, though the plant itself had been cultivated and used since at least the 16th century, introduced by Sephardic Jews who sought refuge in the Ottoman Empire. Court records from 1782 and 1783 shows that chilli pepper served merely for making pickles, though some black African cooks and servants in the palace used chili peppers as seasonings for their okra porridge. Entering the 19th century, chili peppers started to be used by the general populace as seasonings

Corn

Corn is a New World plant, unknown in the Old World before Columbus brought it to Europe in 1493. The new crop spread rapidly: by the 16th century, almost all of southern Europe knew and began to cultivate corn. In the Ottoman Empire, corn maize, appeared first in Croatia in 1611, and spread into the northern Balkans in the following years. Historian Traian Stoianovich also stated that corn was cultivated in Anatolia in at least the 17th century, but we have no further evidence to prove it. The reliable records for corn appeared in the mid-18th century, which call corn “mısır buğdayı” (Egyptian wheat) and that it was cultivated in the district of Galata and Silivri. Since after about 1750, there were maize fields even in the surroundings of the capital, we may conclude that American corn had spread almost all over Anatolia. Corn was eaten by commoners as a cheap food, roasted, boiled, or grilled, and was sold by street vendors on the streets of Istanbul. The elite and palace caught up with corn consumption rather late, the earliest date of corn in the palace was in 1843.

Potato

Compared to other New World vegetables, potatoes entered the Ottoman Empire fairly late. Potatoes began to be planted only in 1835, and the earliest record we have of their use in the palace appeared in the 1850s. Some minuscule amount of potato appeared in the record no more than 7.7 kg on 1851. On 1853, potatoes started to be used regularly every month. In May 1854, some 205 kg of potatoes were shipped to the palace to serve the visiting Prince Napoleon. Further records shows that potatoes were not consumed regularly by the palace, and was only intended to be used for foreign guests. The Melceü’t- Tabbahin defined potato as a kind of Jerusalem artichoke and mentioned it in a recipe which was well known in the French cuisine, and the widespread use of potatoes in cooking only begin in the 20th century with recipes such as potatoes purée becomes popular.