In this work, a leakage-based variant of the Least Mean Fourth (LMF) algorithm, the leaky Least Mean Fourth (LLMF) algorithm, has been derived which helps mitigate the weight drift problem experienced in the conventional LMF algorithm. The main aim of this work is to derive the LLMF adaptive algorithm and conduct transient analysis using the energy conservation relation framework. Finally, a number of simulation results are carried out to corroborate the theoretical findings, and show improved performance obtained through the use of LLMF over the conventional LMF algorithm in a weight drift environment.