We develop a theory of normal-metal -superconductor (NS) and superconductor -superconductor (SS) tunnelling in "bosonic" superconductors with strong attractive correlations taking into account coherence effects in single-particle excitation spectrum and disorder. The theory accounts for the existence of two energy scales, their temperature and doping dependencies, asymmetry and inhomogeneity of tunnelling spectra of underdoped cuprate superconductors.PACS numbers: 74.40.+k, 72.15.Jf, 74.25.Fy Soon after the discovery of high-T c superconductivity [1], a number of tunnelling, photoemission, optical, nuclear spin relaxation and electron-energy-loss spectroscopies discovered an anomalous large gap in cuprate superconductors existing well above the superconducting critical temperature, T c . The gap, now known as the pseudogap, was originally assigned [2] to the binding energy of real-space preformed hole pairs -small bipolarons -bound by a strong electron-phonon interaction (EPI). Present-day scanning tunnelling (STS) [6,7,8], intrinsic tunnelling [9] and angle-resolved photoemission (ARPES) [5] spectroscopies have offered a tremendous advance into the understanding of the pseudogap phenomenon in cuprates and some related compounds. Both extrinsic (see [6,8] and references therein) and intrinsic [9] tunnelling as well as high-resolution ARPES [5] have found another energy scale, reminiscent of a BCSlike "superconducting" gap that opens at T c accompanied by the appearance of Bogoliubov-like quasi-particles [5] around the node. Earlier experiments with a timeresolved pump-probe demonstrated two distinct gaps, one a temperature independent pseudogap and the other a BCS-like gap [10]. Also, Andreev reflection experiments revealed a much smaller gap edge than the bias at the tunnelling conductance maxima in a few underdoped cuprates [11]. Another remarkable observation is the spatial nanoscale inhomogeneity of the pseudogap observed with STS [6,7,8] and presumably related to an unavoidable disorder in doped cuprates, Fig.1a. Essentially, the doping and magnetic field dependence of the superconducting gap compared with the pseudogap and their different real space profiles have prompted an opinion that the pseudogap is detrimental to superconductivity and connected to a quantum critical point rather than to preformed Cooper pairs [9]. Nevertheless without a detailed microscopic theory that can describe highly unusual tunnelling and ARPES spectra, the relationship between the