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Weaken Resolve is not a shooting attack. Can my Battle Psykers use this ability during their shooting phase (thats when the power is used) to effect a unit that is locked in close combat with my platoon? Assuming I win the combat by one or two, it would really make the chance of the enemy unit breaking very high.

As long as its not a PSA, and does not say it is a shooting attack, then you should be able to target a unit in CC, I would have to see the exact wording though to be sure.

it says

The psykers reach out with their minds, evoking irrational terror in the minds of their foes.

This power is used during the pskyer battle squads shooting phase. Choose one unit within 36" and in line of sight of the Psyker battle squad. For the remainder of the turn, the enemy unit's leadership is reduced by the number of Sanctioned Psykers in the unit utilizing the power (to a minimum of 2).

Yes, you may use it on a unit locked in combat. It is not a pshooting attack, and you are not told to "target" but "choose". Target would restrict you, choose does not.