In this paper, we propose a distributed framework for large-scale networks to attain control strategies requiring $k$-hop interactions. This research is motivated by the observation that in many practical applications and operational domains involving large-scale networks, such as environmental monitoring or traffic load balancing, agents may be required to collect only information concerning other agents located sufficiently close to them, that is agents topologically at most $k$-hop away. In this setting, distributed observers available at the state of art, which typically estimates the full network state, may be inadequate due to scalability issues. Differently, we propose a distributed finite-time observer which allows each agent to estimate the state of its $k$-hop neighbors by interacting only with the agents belonging to its 1-hop neighborhood. Furthermore, we demonstrate that for feedback control strategies based on $k$-hop neighborhood information, which are input-to-state stable, the proposed distributed finite-time observer can be effectively used to design stable large-scale networked control strategies. Numerical results are provided to corroborate the theoretical findings.