Learn Farsi: مهره زیرکانه حرکت داده می‌شود.

The word zirak means "clever" or "shrewd". Adding the suffix -âne turns it into the adverb zirakāne, meaning "cleverly" or "in a clever manner." This suffix is a very common way to create adverbs of manner that describe how an action is performed.

Why is the verb harkat dāde mishavad instead of just harkat mikonad?

Because the sentence is in the passive voice: the piece is moved by a player, it isn't moving on its own. To make a compound verb like harkat dādan (to move something) passive, you change the verbal part into a past participle (dāde) and add the auxiliary verb shodan (here, conjugated as the present tense mi-shavad).

What kind of "piece" does mohre refer to in this context?

Mohre refers specifically to a playing piece in board games like chess, checkers, or backgammon. You wouldn't use it for a piece of cake (which is tekke) or a piece of paper. Outside of games, mohre can also mean a bead or a small nut (as in nuts and bolts).

Is the final letter of zirakāne pronounced as an "h"?

No, the final ه (heh) in zirakāne makes a short "e" sound, not an "h" sound. The suffix -âne is always pronounced with this final "e", which is a classic marker of these adverbs in Farsi.