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A transverse kick resulting from the coaxial rf coupler in the L-band rf gun at the photoinjector test facility at DESY in Zeuthen (PITZ) has been systematically studied in simulations and further characterized in experiments. The used rf macro-pulse for the gun is typically 600 μs long and applied to produce a train of up to 2700 electron bunches. The kick is transient and found to be dependent on the detuning of the resonance frequency of the gun cavity. The frequency detuning within the rf pulse results in a variation in the kick strength along the pulse. This leads to a downstream orbit and size change of individual bunches within the train. Using 3D rf field distributions, calculated at resonant and detuned frequencies of the cavity, particle tracking simulations are performed to evaluate the integral kick location and strength, simulate its transient behaviors with respect to gun launch phase and detuned frequency of the gun cavity, and thereby emulate its impacts onto the electron bunch. In the experiments, the temperature of the cooling water for the gun is tuned, allowing detailed characterization of the frequency detuning within the rf pulse and thus measurements of the kick which are of practical interest for machine operation. The resulting orbit and size change along a 240 μs bunch train have been measured downstream at the injector exit for an rf gun peak power of about 5.7 MW. A measured focusing difference between head and tail of the bunch train is evaluated in terms of the gun solenoid current which provides more relevant information for overall tuning of the FELs. This is exemplarily given for both the under- and over-focusing case. The obtained results will be presented and discussed.