In Federated Learning (FL), devices-also referred to as clients-can exhibit heterogeneous availability patterns, often correlated over time and with other clients. This paper addresses the problem of heterogeneous and correlated client availability in FL. Our theoretical analysis is the first to demonstrate the negative impact of correlation on FL algorithms' convergence rate and highlights a trade-off between optimization error (related to convergence speed) and bias error (indicative of model quality). To optimize this trade-off, we propose Correlation-Aware FL (CA-Fed), a novel algorithm that dynamically balances the competing objectives of fast convergence and minimal model bias. CA-Fed achieves this by dynamically adjusting the aggregation weight assigned to each client and selectively excluding clients with high temporal correlation and low availability. Experimental evaluations on diverse datasets demonstrate the effectiveness of CA-Fed compared to state-of-the-art methods. Specifically, CA-Fed achieves the best trade-off between training time and test accuracy. By dynamically handling clients with high temporal correlation and low availability, CA-Fed emerges as a promising solution to mitigate the detrimental impact of correlated client availability in FL.