The potential application of artificial spin ice in magnetic nanodevices provides a strong drive to investigate different lattice geometries. Here, we combine components of a recently investigated artificial spin ratchet with components of the prototypical square lattice to form a geometrically frustrated artificial spin ice system where Ising-type nanomagnets are arranged onto a two-dimensional square-kite lattice. Using synchrotron-based photoemission electron microscopy, we explore moment configurations achieved in this lattice geometry. Following thermal annealing, we image how a variation of the relevant lattice parameter affects magnetic ordering in four-island squares and four-island vertices during cooling through the Blocking temperature. Depending on lattice spacing, both nearly uniform and disordered spin configurations are accessible in our samples. We show that the relative energies of the building blocks of the system, which are typically used to classify lattice configurations, are not predictive of the low energy states adopted by the experimental system. To understand magnetic ordering in the square-kite lattice, longer range interactions must be considered.