  In this work, we study a chemotaxis-Navier-Stokes model in a two-dimensional
setting as below,
  \begin{eqnarray}
  \left\{
  \begin{array}{llll}
  \displaystyle n_{t}+\mathbf{u}\cdot\nabla n=\Delta n-\nabla \cdot(n\nabla
c)+f(n), &&x\in\Omega,\,t>0,\\
  \displaystyle c_{t}+\mathbf{u}\cdot\nabla c=\Delta c - c+ n,
&&x\in\Omega,\,t>0,\\
  \displaystyle \mathbf{u}_{t}+\kappa(\mathbf{u}\cdot\nabla)\mathbf{u}=\Delta
\mathbf{u} +\nabla P+ n\nabla\phi, &&x\in\Omega,\,t>0,\\
  \displaystyle \nabla\cdot\mathbf{u}=0,&&x\in\Omega,\,t>0.\\
  \end{array}
  \right. \end{eqnarray} Motivated by a recent work due to Winkler, we aim at
investigating generalized solvability for the model the without imposing a
critical superlinear exponent restriction on the logistic source function $f$.
Specifically, it is proven in the present work that there exists a triple of
integrable functions $(n,c,\mathbf{u})$ solving the system globally in a
generalized sense provided that $f\in C^1([0,\infty))$ satisfies $f(0)\ge0$ and
$f(n)\le rn-\mu n^{\gamma}$ ($n\ge0$) with any $\gamma>1$. Our result indicates
that persistent Dirac-type singularities can be ruled out in our model under
the aforementioned mild assumption on $f$. After giving the existence result
for the system, we also show that the generalized solution exhibits eventual
smoothness as long as $\mu/r$ is sufficiently large.
