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The BaBar, Belle, and LHCb collaborations have reported evidence for newphysics in $B\to D\tau\nu$ and $B\to D^*\tau\nu$ of approximately $3.8\sigma$.There is also the long lasting discrepancy of about $3\sigma$ in the anomalousmagnetic moment of the muon, and the branching ratio for $\tau\to\mu\nu\nu$ is$1.8\sigma$ ($2.4\sigma$) above the Standard Model expectation using the HFAG(PDG) values. Furthermore, CMS found hints for a non-zero decay rate of$h\to\mu\tau$. Interestingly, all these observations can be explained byintroducing new scalars. In this article we consider these processes within alepton-specific two-Higgs doublet model (i.e. of type X) with additionalnon-standard Yukawa couplings. It is found that one can accommodate$\tau\to\mu\nu\nu$ with modified Higgs--$\tau$ couplings. The anomalousmagnetic moment of the muon can be explained if the additional neutral CP-evenHiggs $H$ is light (below 100 GeV). Also $R(D)$ and $R(D^*)$ can be easilyexplained by additional $t$--$c$--Higgs couplings. Combining these $t$--$c$couplings with a light $H$ the decay rate for $t\to H c$ can be in a testablerange for the LHC. Effects in $h\to\mu\tau$ are also possible, but in this casea simultaneous explanation of the anomalous magnetic moment of the muon isdifficult due to the unavoidable $\tau\to\mu\gamma$ decay.

A perturbed lepton-specific two-Higgs-doublet model facing experimental hints for physics beyond the Standard Model