Recently the CDF collaboration reported a new measurement of the W boson mass using their full data set of 8.8 fb −1 in pp collision at CDF detector (Fermilab Tevatron), which deviates from the Standard Model (SM) prediction about 7σ. By including this new measurement, however, we explore the implications in the framework of general two Higgs doublet model type-III in presence of a specific Yukawa texture, wherein Lepton Flavour Violating (LFV) (neutral) currents are induced at tree level. The paper shows the preferred values of oblique parameters S and T which is simultaneously consistent with the 96 GeV excesses observed by LEP and CMS at about 2.8σ When combined with current LHC data and electroweak precision measurements, we show that the parameter space that accommodates the excesses can also significantly contribute to M W disfavors degenerates charged Higgs mass with neutral CP-odd Higgs when H being Standard Model like particle. We show that 96 GeV excesses may persist only in the case m H ± > m A . Both Branching ratios of H ± and A are strictly constrained.