Can I use an AND gate before a clock input?
The naming scheme on the primitives in your image suggests that this design is intended to be implemented in an FPGA.

If this is the case, gating a clock network is not recommended. Clocked logic elements in an FPGA can usually only be clocked by dedicated clock networks. These networks are only available in very limited numbers (i.e, perhaps a dozen on the entire chip!). Instantiating logic which passes a signal from combinational logic to a clock will typically result in one of these networks being used for that signal, making it unavailable for other logic on the chip.

To avoid this, most FPGAs will make logic primitives available with separate combinational logic "enable" signals. (Xilinx calls theirs FDCE/FDPE/FDRE/FDSE depending on the state at reset, and Altera calls theirs DFFE.) These logic elements will (more or less) AND the clock signal with the combinational enable signal, giving the same result as your schematic here without the detrimental effects on clock networks.