We use ultrafast optical spectroscopy to study the nonequilibrium quasiparticle relaxation dynamics of the iron-based superconductor KCa$_2$Fe$_4$As$_4$F$_2$ with $T_c=33.5$ K. Our results reveal an evident pseudogap ($Delta_{PG}$ = 2.4 $pm$ 0.1 meV) below $T^*approx 50$ K but prior to the opening of a superconducting gap ($Delta_{SC}$(0) $approx$ 4.3 $pm$ 0.1 meV). Measurements under high pump fluence real two distinct coherent phonon oscillations with frequencies of 1.95 and 5.51 THz, respectively. The high-frequency mode corresponds to the $c-$axis polarized vibrations of As atoms ($A_{1g}$ mode) with a nominal electron-phonon coupling constant $lambda_{A_{1g}}$ = 0.194 $pm$ 0.02. Below $T_c$, the temperature dependence of both frequency and damping rate of $A_{1g}$ mode clearly deviate from the description of optical phonon anharmonic effects. These results suggest that the pseudogap is very likely a precursor of superconductivity, and the electron-phonon coupling may play an essential role in the superconducting pairing in KCa$_2$Fe$_4$As$_4$F$_2$.
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