Favoritism or "playing favorites" was preferential treatment given to a certain individual based on one's admiration of another, rather than individual merit. To "curry favor" was to seek favoritism through flattery.
In 2258 of the alternate reality, Spock told Nyota Uhura that he assigned her to the USS Farragut instead of the USS Enterprise to avoid the appearance of favoritism. (Star Trek)
According to Spock, Starbase Yorktown was built in open space to avoid the appearance of geographical favoritism for any Federation member worlds. (Star Trek Beyond)
Tom Paris received a B minus in his Survival Strategies course at Starfleet Academy. When he related this to Neelix in 2372, adding that his father was teaching the course at the time, the Talaxian was surprised, commenting that the elder Paris didn't play favorites. (VOY: "Parturition")