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We assess the performance of the multipole expansion formalism in the case of single-dish HI intensity mapping, including instrumental and foreground removal effects. This formalism is used to provide MCMC forecasts for a range of HI and cosmological parameters, including redshift space distortions and the Alcock-Paczynski effect. We first determine the range of validity of our power spectrum modelling by fitting to simulation data, concentrating on the monopole, quadrupole, and hexadecapole contributions. We then show that foreground subtraction effects can lead to severe biases in the determination of cosmological parameters, in particular the parameters relating to the transverse BAO rescaling, the growth rate and the HI bias ($α_\perp$, $\overline{T}_\text{HI} fσ_8$, and $\overline{T}_\text{HI} b_\text{HI} σ_8$, respectively). We attempt to account for these biases by constructing a 2-parameter foreground modelling prescription, and find that our prescription leads to unbiased parameter estimation at the expense of increasing the estimated uncertainties on cosmological parameters. In addition, we confirm that instrumental and foreground removal effects significantly impact the theoretical covariance matrix, and cause the covariance between different multipoles to become non-negligible. Finally, we show the effect of including higher-order multipoles in our analysis, and how these can be used to investigate the presence of instrumental and systematic effects in HI intensity mapping data.