Data analysis from upcoming large galaxy redshift surveys, such as Euclid and DESI, will significantly improve constraints on cosmological parameters. To optimally extract the maximum information from these galaxy surveys, it is important to control with a high level of confidence the uncertainty and bias arising from the estimation of the covariance that affects the inference of cosmological parameters. In this work, we address two different but closely related issues: (i) the sampling noise present in a covariance matrix estimated from a finite set of simulations and (ii) the impact on cosmological constraints of the non-Gaussian contribution to the covariance matrix of the power spectrum. We focussed on the parameter estimation obtained from fitting the full shape of the matter power spectrum in real space, using the Dark Energy and Massive Neutrino Universe (DEMNUni) N-body simulations. Parameter inference was done through Monte Carlo Markov chains. Regarding the first issue, we adopted two different approaches to reduce the sampling noise in the precision matrix that propagates in the parameter space: on the one hand, using an alternative estimator of the covariance matrix based on a non-linear shrinkage, NERCOME (which stands for Non-parametric Eigenvalue-Regularised COvariance Matrix Estimator); and, on the other hand, employing a method of fast generation of approximate mock catalogues, COVMOS. We find that NERCOME can significantly reduce the stochastic shifts of the posteriors of parameters, but at the cost of a systematic overestimation of the error bars on the cosmological parameters. We show that using a COVMOS covariance matrix estimated from a large number of realisations (10 000) results in unbiased cosmological constraints. Regarding the second issue, we quantified the impact on cosmological constraints of the non-Gaussian part of the power spectrum covariance purely coming from non-linear clustering. We find that when this term is neglected, both the uncertainties and best-fit values of the estimated parameters are affected for a scale cut kmax > 0.2 h/Mpc.