  Recently, a weak converse theorem for Borcherds' lifting operator of type
$O(2,1)$ for $\G_0(N)$ is proved and the logarithmic derivative of a modular
form for $\G_0(N)$ is explicitly described in terms of the values of
Niebur-Poincar\'e series at its divisors in the complex upper half-plane. In
this paper, we prove that the generalized Borcherds' lifting operator of type
$O(2,1)$ is Hecke equivariant under the extension of Guerzhoy's multiplicative
Hecke operator on the integral weight meromorphic modular forms and the Hecke
operator on half-integral weight vector-valued harmonic weak Maass forms.
Additionally, we show that the logarithmic differential operator is also Hecke
equivariant under the multiplicative Hecke operator and the Hecke operator on
integral weight meromorphic modular forms. As applications of Hecke
equivariance of the two operators, we obtain relations for twisted traces of
singular moduli modulo prime powers and congruences for twisted class numbers
modulo primes, including those associated to genus $1$ modular curves.
