The caster level increase affects all spells the master specialist casts of a particular school of magic no matter the spell's source
The prestige class master specialist (Complete Mage 70–3) has as entry requirements both Spellcasting: Must be able to cast 2nd-level arcane spells and Special: Must be a specialist wizard. Assuming a multiclass cleric/master specialist somehow meets these requirements—probably by also taking levels in the wizard class—, the level 6 and 9 master specialist class feature caster level increase has no restrictions on the nature of the spells it affects, saying
Upon reaching 6th level, add 1 to your caster level whenever you cast a spell of your chosen school. At 9th level, you instead add 2 to your caster level. (71)
The special ability does not, for instance, limit the increased caster level only to wizard spells the creature casts. Thus, for example, a cleric 1/conjurer 3/master specialist (conjuration) 6 would cast wizard conjuration spells as if he were a level 10 wizard and cleric conjuration spells as if he were a level 2 cleric. (Note that this doesn't change the sheer number or the spell levels of cleric or wizard spells such a creature can cast. Those remains governed almost exclusively by the creature's effective cleric and wizard class levels, respectively.)
Note: In optimization circles it's typically considered a bad idea to lose any caster levels once a casting class is picked, but a 1-level dip in cleric is often one the handful of such losses that can still be worthwhile to the right character despite a loss. However, the bonus to caster level to one school of spells due to the prestige class master specialist's special ability caster level increase in no way makes up for the example creature's otherwise minimal cleric casting; that creature likely took a level of cleric for the domain granted powers, the turning, the expansive spell list, or almost anything else besides the actual casting!