The beta decay of 50K and 38Ca has been investigated with the main motive of determining more accurately the first-forbidden beta branches, in particular the rank-0, Delta J=0, beta transitions. 50K and 38Ca have been produced by fragmentation of U and Ti targets, respectively, with a 1 GeV proton beam and subsequent on-line mass separation. For 50K, gamma -ray spectroscopy, as well as delayed neutron spectroscopy by time of flight, was carried out to obtain a detailed decay scheme to 20 (bound and unbound) levels in 50Ca. The level structure of 50Ca can be compared to recent calculations which incorporate one-particle–one-hole excitations from the f7/2 shell. The first-forbidden beta - transition 50K(0-)--> 50Ca(0+) ground state has been evaluated for the first time by a direct measurement of beta and gamma activities. Its strength (61.0±7.4%) is interpreted as an effect of the meson-exchange current (MEC) leading to an enhancement factor of 62(5)% in comparison with the value predicted by shell-model calculations using the impulse approximation. For the 38Ca-->38K decay, chemical selective production was obtained through separation of the molecular ion CaF+ without contamination by isobars. In these conditions, the measurement of very weak beta branches, at a level of 10-3 per 100 decays, could be made and a limit, at the 2 sigma confidence level, has been obtained for the 0+-->0- branch to the level at Ex=2993 keV (I beta <0.0046%). Implications of these results on the general trend of meson-exchange enhancements of first-forbidden transitions within the framework of the spherical shell model are discussed.