You don't normally include the grounds into your cup, but in the case of Turkish ground coffee, sometimes it's a whole different story. Because of how this coffee is brewed–which doesn't involve any filters–pouring into your serving vessel will most likely include grounds.
However, it's important to note that Turkish coffee uses very finely ground coffee beans (Turkish grind setting), thus allowing some to dissolve in the water, but not totally. And this coffee is also intended to be drunk slowly to ensure that every sweet sip is enjoyable. In this manner, you will also be more cautious of how the taste and consistency become. If you think it's already too bitter and too thick to consume, then you can leave the cup unfinished that way.