There are multiple reasons why a city may have a fortified Citadel. To illustrate, I'm going to give two examples from very different times and places. Antioch during the First Crusade and Barcelona in the 19th Century.

During the First Crusade, Christian crusaders succeeded in conquering the city of Antioch after multiple sieges. However, they were unable to take the citadel of the city. The Muslim garrison of the citadel held out, forcing the crusaders to expend resources to continue defending them. The garrison only surrendered when a Muslim relief force was defeated. Here, the citadel served as a place where defenders could hold out to expend enemy resources and provide assistance to a relief force.

Relevant quotes:

"In addition, let all know, also, that I have the Franks shut up in Antioch, and that I hold the citadel at my free disposal, while they (the enemy) are below in the city."

"Count of St. Gilles, who remained behind to watch the citadel for fear lest the Turks descend into the city" (Gesta)

"For the Turks who were guarding the citadel of the city gave up hope upon seeing the headlong flight of their people; some, on the Pledge of their lives alone, surrendered themselves to us, and the rest fled headlong." (Raymond d'Aguiliers)

In Barcelona, since at least 1800, Montjuïc Castle has served as a symbol of Royal authority over the city of Barcelona and the region of Catalonia. For example, "In December 1842, the Spanish regent Joaquín Baldomero Fernández Espartero Álvarez de Toro bombarded the city of Barcelona to strike down a revolt against his free trade policy." Additionally, the castle has served as a location for the execution of numerous Catalan "Nationalists" (I'm using the term Nationalist loosely). In this case, the citadel serves as a symbol and power base for a distant authority to control the larger city.

Antioch Source: http://sourcebooks.fordham.edu/halsall/source/cde-antioch.asp

Barcelona Source: https://barcelona.de/en/barcelona-castell-montjuic.html