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The quasi-brittle nature of rocks challenges the basic assumptions of linear hydraulic fracture mechanics (LHFM): linear elastic fracture mechanics and smooth parallel plates lubrication fluid flow. We relax these hypotheses and investigate the growth of a plane-strain hydraulic fracture in an impermeable medium accounting for a rough cohesive zone and a fluid lag. In addition to a dimensionless toughness and the time-scale of coalescence of the fluid and fracture fronts as in the LHFM case, the solution now also depends on the in-situ-to-cohesive stress ratio and the intensity of the flow deviation induced by aperture roughness. The solution is appropriately described by a nucleation time-scale, which delineates the fracture growth into a nucleation phase, an intermediate stage and a late time stage where convergence toward LHFM predictions finally occurs. A highly non-linear hydro-mechanical coupling takes place as the fluid front enters the rough cohesive zone which itself evolves during the nucleation and intermediate stages. This coupling leads to significant additional viscous flow dissipation. As a result, the fracture evolution deviates from LHFM solutions with shorter fracture lengths, larger widths and net pressures. These deviations ultimately decrease at late times as the lag and cohesive zone fractions both become smaller. The deviations increase with larger dimensionless toughness and in-situ-to-cohesive stress ratio, as both further localize viscous dissipation near the fluid front located in the rough cohesive zone. The convergence toward LHFM can occur at very late time for realistic values of in-situ-to-cohesive stress ratio encountered at depth. The impact of a rough cohesive zone appears to be prominent for laboratory experiments and short in-situ injections in quasi-brittle rocks with ultimately a larger energy demand compared to LHFM predictions.