What is distorted square pyramidal structure of a compound?
Going by a standard argumentation, the $\ce{[XeF5]+}$ cation would be expected to have two different bond types: a classical 2-electron-2-centre bond to the axial fluorine and a pair of two 4-electron-3-centre bonds making up the base of the pyramid with the equatorial fluorines. This structure is shown in figure 1.


Figure 1: Hypothetical perfect structure of $\ce{[XeF5]+}$. Image taken from Wikipedia, where a full list of authors is available.

The actual structure of $\ce{[XeF5]+}$, reported by Leary et al.,[1] is not perfect, though. The axial fluorine atom does not sit at the centre of the pyramid, destroying the perfect $C_\mathrm{4v}$ point group. This is probably an effect of core orbitals on the central xenon atom (I cannot access the paper where I am right now). Because the pyramid no longer has four equal faces, it is distorted.

Reference:
K. Leary, D. H. Templeton, A. Zalkin, N. Bartlett, Inorg. Chem. 1973, 12, 1726. DOI: 10.1021/ic50126a004.