We analyze the higher-twist effects and the $SU(3)$-flavour symmetry breaking
in the correlation functions used to calculate form factors of
pseudoscalar mesons in the QCD light-cone sum rule approach.
It is shown that the Ward identities for these correlation functions yield
relations between twist-4 two- and three-particle distribution
amplitudes. In addition to the relations already obtained from the QCD
equations of motions, we have found a new one. With the help of these
relations, the twist-4 contribution to the light-cone sum rule
for the pion electromagnetic form factor is reduced to a very simple
form. Simultaneously, we correct a sign error in the earlier calculation.
The updated light-cone sum rule prediction for the pion
form factor at intermediate momentum transfers is compared with the recent
Jefferson Lab data. Furthermore, from the correlation functions with
strange-quark currents the kaon electromagnetic form factor and
the $K\to \pi$ weak transition form factors are predicted with
$O(m_s)\sim O(m_K^2)$ accuracy.