  We point out that chiral condensates at nonzero temperature and magnetic
fields are in strict connection to the space-time integral of corresponding
two-point neutral meson correlation functions in the pseudoscalar channel via
the Ward-Takahashi identity. Screening masses of neutral pseudoscalar mesons,
which are defined as the exponential decay of the corresponding spatial
correlation functions in the long distance, thus are intrinsically connected to
(inverse) magnetic catalysis of chiral condensates. To study this we performed
lattice simulations of $(2+1)$-flavor QCD on $32^3\times N_t$ lattices with
pion mass $M_\pi\simeq 220$ MeV in a fixed scale approach having temperature
$T\in[17, 281]$ MeV and magnetic field strength $eB\in[0, 2.5]$ GeV$^2$. We
find that screening lengths, i.e. inverses of screening masses of $\pi^0$,
$K^0$ and $\eta^0_{s\bar{s}}$, turn out to have the similar complex $eB$ and
$T$ dependences of the corresponding chiral condensates. Although the
transition temperature is found to always decrease as $eB$ grows, we show that
the suppression due to magnetic fields becomes less significant for hadron
screening length and chiral condensates with heavier quarks involved, and
ceases to occur for $\eta^0_{s\bar{s}}$ and strange quark chiral condensate.
The complex $eB$ and $T$ dependences of both screening masses and chiral
condensates, reflecting the crossover nature of the QCD transition, are
attributed to the competition between sea and valence quark effects. These
findings could be useful to guide low-energy models and effective theories of
QCD.
