  We theoretically investigate a supersymmetric collective mode called
Goldstino in a Bose-Fermi mixture. The explicit supersymmetry breaking, which
is unavoidable in cold atom experiments, is considered. We derive the
Gell-Mann--Oakes-Renner (GOR) relation for the Goldstino, which gives the
relation between the energy gap at the zero momentum and the explicit breaking
term. We also numerically evaluate the gap of Goldstino above the Bose-Einstein
condensation temperature within the random phase approximation (RPA). While the
gap obtained from the GOR relation coincides with that in the RPA for the
mass-balanced system, there is a deviation from the GOR relation in the
mass-imbalanced system. We point out the deviation becomes large when the
Goldstino pole is close to the branch point, although it is parametrically a
higher order with respect to the mass-imbalanced parameter. To examine the
existence of the goldstino pole in realistic cold atomic systems, we show how
the mass-imbalance effect appears in $^6$Li-$^7$Li, $^{40}$K-$^{41}$K, and
$^{173}$Yb-$^{174}$Yb mixtures. Furthermore, we analyze the Goldstino spectral
weight in a $^{173}$Yb-$^{174}$Yb mixture with realistic interactions and show
a clear peak due to the Goldstino pole. As a possibility to observe the
Goldstino spectrum in cold atom experiments, we discuss the effects of the
Goldstino pole on the fermionic single-particle excitation as well as the
relationship between the GOR relation and Tan's contact.
