The cold dark matter (CDM) scenario has proved successful in cosmology.However, we lack a fundamental understanding of its microscopic nature.Moreover, the apparent disagreement between CDM predictions andsubgalactic-structure observations has prompted the debate about its behaviourat small scales. These problems could be alleviated if the dark matter iscomposed of ultralight fields $m \sim 10^{-22}\ \text{eV}$, usually known asfuzzy dark matter (FDM). Some specific models, with axion-like potentials, havebeen thoroughly studied and are collectively referred to as ultralight axions(ULAs) or axion-like particles (ALPs). In this work we consider anharmoniccorrections to the mass term coming from a repulsive quartic self-interaction.Whenever this anharmonic term dominates, the field behaves as radiation insteadof cold matter, modifying the time of matter-radiation equality. Additionally,even for high masses, i.e. masses that reproduce the cold matter behaviour, thepresence of anharmonic terms introduce a cut-off in the matter power spectrumthrough its contribution to the sound speed. We analyze the model and deriveconstraints using a modified version of CLASS and comparing with CMB andlarge-scale structure data.