Suppose we start with the line bundle $ O(N) \rightarrow \mathbb{C} \mathbb{P}^m$. By a holomorphic connection, I mean an operator $\nabla : \Gamma(\mathbb{C} \mathbb{P}^m, O(N)) \rightarrow \Gamma(\mathbb{C} \mathbb{P}^m, O(N)) \otimes \Omega^1(\mathbb{C} \mathbb{P}^m)$ where $\Omega^1(\mathbb{C} \mathbb{P}^m)$ is the set of holomorphic 1-forms. By meromorphic connection, I mean that we replace $\Omega^1(\mathbb{C} \mathbb{P}^m)$ with $\Omega^1_M(D)$,the set of meromorphic one-forms with poles in some prescribed polar divisor $D$.
I have two questions about these things:
From my understanding, there does not exist a holomorphic connection for our line bundle, since such a connection would be automatically flat, and thus our line bundle would be flat, which is false for $N \neq 0$. Am I correct in this assertion?
Do there exists meromorphic connections? If so, can one provide an explicit example?
The answer to both your questions is positive.
As is explained in Huybrechts's "Complex Geometry – An Introduction" here, the existence of a holomorphic connection is equivalent to the Atiyah class being trivial.
Huybrechts, Daniel. Complex geometry. An introduction. Universitext. Springer-Verlag, Berlin, 2005. xii+309 pp.
The Atiyah class in this case is just the first Chern class which is nontrivial for $N \ne 0$.
Just trivialize $\mathcal{O}(N)$ on a standard open set. Any connection there can be considered as a meromorphic connection in your sense.