What do müdrik, mesbûk, and lâhik mean? How do these individuals perform their prayers?

In the dictionary, müdrik means “one who has caught up, reached, or joined.” As a religious term, it refers to a person who joins the imam in prayer at the latest during the bowing (ruku‘) of the first rak‘ah and performs the prayer together with the imam.

Lâhik refers to someone who starts the prayer with the imam but, due to an interruption such as losing their ablution during the prayer, is forced to pause and thus cannot perform part of the prayer with the imam. If this person knows the proper procedure, after renewing their ablution, they rejoin the congregation and continue their prayer. After the imam gives the final salutation, they complete the rak‘ahs they missed. If they do not know the proper procedure, they must repeat the prayer from the beginning.

Mesbûk is someone who joins a congregational prayer late, that is, after the bowing (ruku‘) of the first rak‘ah. A person who does not join the imam in ruku‘ for at least the time it takes to say “subhanallah” is considered to have missed that rak‘ah. When the imam gives the final salutation, the mesbûk stands up without delay (unless a prostration of forgetfulness is required) and completes the rak‘ahs missed with the congregation on their own. While making up these missed rak‘ahs, the mesbûk is like someone praying alone: in the rak‘ahs where one would recite a surah or verse after al-Fatiha when praying alone, they do so; in those where it is not recited, they do not. (el-Fetâva’l-Hindiyye, 1/91-92; Ibn ‘Abidin, Reddü’l-muhtâr, 1/594)

Source: T.C. Diyanet İşleri Başkanlığı, Din İşleri Yüksek Kurulu