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What do we mean by the statement when we say that a cavity mode is driven by a classical field or classically controlled field, in Cavity Optomechanics?

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It means that one or multiple channel modes (an external mode in the Jaynes-Cummings Model of CQED) of the cavity are driven by a coherent state. Here "driven by a coherent state" is to be understood within the Input-Output formalism: The input operator is in a coherent state.

It turns out that for a coherent state the quantum stochastic equations of motion can be replaced by an effective Hamiltonian that is equivalent to a classical driving field Hamiltonian.

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