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(Abridged) Very high-energy (VHE, $E>100$ GeV) observations of the blazar Mrk501 with MAGIC in 2014 have provided evidence for an unusual narrow spectralfeature at about 3 TeV during an extreme X-ray flaring activity. The one-zonesynchrotron-self Compton scenario, widely used in blazar broadband spectralmodeling, fails to explain the narrow TeV component. Motivated by this rareobservation, we propose an alternative model for the production of narrowfeatures in the VHE spectra of flaring blazars. These spectral features mayresult from the decay of neutral pions ($\pi^0$ bumps) that are in turnproduced via interactions of protons (of tens of TeV energy) with energeticphotons, whose density increases during hard X-ray flares. We explore theconditions needed for the emergence of narrow $\pi^0$ bumps in VHE blazarspectra during X-ray flares reaching synchrotron energies $\sim100$ keV usingtime-dependent radiative transfer calculations. We focus on high-synchrotronpeaked (HSP) blazars, which consist the majority of VHE detected extragalacticsources. We find that synchrotron-dominated flares with peak energies$\gtrsim100$ keV can be ideal periods for the search of $\pi^0$ bumps in theVHE spectra of HSP blazars. Application of the model to the SED of Mrk 501 onMJD 56857.98 shows that the VHE spectrum of the flare is described well by thesum of an SSC component and a distinct $\pi^0$ bump centered at 3 TeV. Spectralfitting of simulated SSC+$\pi^0$ spectra for the Cherenkov Telescope Array(CTA) show that a $\pi^0$ bump could be detected at a 5$\sigma$ significancelevel with a 30-min exposure.

TeV pion bumps in the gamma-ray spectra of flaring blazars