Pulling-Back a Current by a Holomorphic Proper Non-Submersion Function Between Two Compact Complex Surfaces

Question

Let $M,N$ be two Compact Complex Surfaces (compact complex manifolds of complex dimension $2$).

Let $A$ be a (non-empty) subset of $M$ (not necessarily a sub-manifold).

Let $f: M \longrightarrow N$ be a map satisfying the following conditions:

1. $f$ is Holomorphic;
2. $f$ is Proper (the inverse image of a compact set is compact as well);
3. $f$ is not, unfortunately, Submersion;
4. $f(A)$ is a Finite subset of $N$;
5. $f: M \setminus A \longrightarrow N\setminus f(A)$ is a Diffeomorphism (or even a Bi-Holomorphic).

Let $T$ be a (non-trivial) $(1,1)$ Current on the surface $N$.

Under the assumption above

Can the current $T$ (on the surface $N$) be pulled-back by the map $f$ to a current on the surface $M$?

The third condition (the map $f$ is not submersion) has, indeed, a non-trivial crucial effect. Otherwise, it is possible to define a current $T^*$ on the surface $M$ as follows $$T^*(\theta):=T(f_*(\theta))$$ where $\theta$ is a $(1,1)$ differential form on the surface $M$.