Numerical simulations of observed large-scale structure formation work best with Cold Dark Matter (CDM; see the answer here). Neutrinos are candidates for Hot Dark Matter (HDM), and hence they cannot account for the total dark matter abundance of the Universe. By the same token, axions are also relativistic because they have very tiny masses. Aren't they also candidates of HDM like neutrinos? Shouldn't they also be disfavoured for the same reason?
Answer
The answer is that the axions are not relativistic, but rather extremely cold. Neutrinos are hot because they were in thermal equilibrium with the standard model heat bath before they decoupled.
This is not the case for axions, which needs some form of non-thermal production mechanism. Otherwise they could only form hot dark matter, as you say. If the axions wre thermally produced their relic density would also be far too low to explain dark matter (even if hot dark matter was not ruled out).
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