Adsorbed transition metal atoms can have partially filled d or f shells due to strong on-site Coulomb interaction. Capturing all effects originating from electron correlation in such strongly correlated systems is a challenge for electronic structure methods. It requires a sufficiently accurate description of the atomistic structure (in particular bond distances and angles), which is usually obtained from first-principles Kohn-Sham density functional theory (DFT), which due to the approximate nature of the exchange-correlation functional may provide an unreliable description of strongly correlated systems. To elucidate the consequences of this popular procedure, we apply a combination of DFT with the Anderson impurity model (AIM), as well as DFT + U for a calculation of the potential energy surface along the Co/Cu(001) adsorption coordinate, and compare the results with those obtained from DFT. The adsorption minimum is shifted towards larger distances by applying DFT+AIM, or the much cheaper DFT+U method, compared to the corresponding spin-polarized DFT results, by a magnitude comparable to variations between different approximate exchange-correlation functionals (0.08 to 0.12 Å). This shift originates from an increasing correlation energy at larger adsorption distances, which can be traced back to the Co 3dxy and 3dz2 orbitals being more correlated as the adsorption distance is increased. We can show that such considerations are important, as they may strongly affect electronic properties such as the Kondo temperature.