For any
n
∈
N
=
{
0
,
1
,
2
,
…
}
n\in \mathbb {N}=\{0,1,2,\ldots \}
and
b
,
c
∈
Z
b,c\in \mathbb {Z}
, the generalized central trinomial coefficient
T
n
(
b
,
c
)
T_n(b,c)
denotes the coefficient of
x
n
x^n
in the expansion of
(
x
2
+
b
x
+
c
)
n
(x^2+bx+c)^n
. Let
p
p
be an odd prime. In this paper, we determine the summations
∑
k
=
0
p
−
1
T
k
(
b
,
c
)
2
/
m
k
\sum _{k=0}^{p-1}T_k(b,c)^2/m^k
modulo
p
2
p^2
for integers
m
m
with certain restrictions. As applications, we confirm some conjectural congruences of Sun [Sci. China Math. 57 (2014), pp. 1375–1400].