Motivated by the challenge of maximizing connected coverage with limited UAVs in communication networks, we address the problem within a graph network framework \(G=(V,E)\) , where V represents potential UAV deployment positions and E denotes communication links between nodes. A utility function \(f: 2^{V}\rightarrow \mathbb {R}_{+}\) is defined to characterize coverage efficiency. Under the constraint of limited field-of-view (FoV) UAVs, the objective is to identify a subset \(S\subseteq V\) with \(|S|\le K\) that maximizes f(S) while ensuring the induced subgraph G[S] remains connected. We formulate this as the Maximum Connected Submodular function with Cardinality constraint (MCSC) problem and propose a \(\frac{1-e^{-1}}{2\sqrt{K-1}+5}\) -approximation algorithm, leveraging a novel tree decomposition technique. Additionally, we present a bicriteria \(\left( \frac{(1-e^{-1})\alpha }{2\sqrt{K}+3\alpha },\alpha ^2\right) \) -approximation algorithm for the problem, where \(\alpha >1\) is a constant. For a special case of the MCSC problem, where the submodular utility exhibits partial additivity when subsets are sufficiently far apart, we define the Maximum Connected h-Hop Submodular function with a Cardinality constraint (MCHSC) problem. We provide an approximation algorithm with a ratio of \((1-2\varepsilon )\left( \frac{1-e^{-1}}{5(h+1)+1}-\delta \right) \) when \(K > 25h(h+1) -5 \) , where \(\varepsilon \) , \(\delta \) are small positive constants and h captures the partial additivity property.