We consider slowly time-dependent singular stochastic partial differential equations on the two-dimensional torus, driven by weak space-time white noise, and renormalised in the Wick sense. Our main results are concentration results on sample paths near stable equilibrium branches of the equation without noise, measured in appropriate Besov and H\"older norms. We also discuss a case involving a pitchfork bifurcation. These results extend to the two-dimensional torus those obtained in [Berglund and Gentz, PTRF 2002] for finite-dimensional SDEs, and in [Berglund and Nader, Stochastics and Partial Differential Equations, 2022] for SPDEs on the one-dimensional torus.