  Nilpotent BRST operators for higher-spin $W_{2,s}$ strings, with currents of
spins 2 and $s$, have recently been constructed for $s=4$, 5 and 6. In the case
of $W_{2,4}$, this operator can be understood as being the BRST operator for
the critical $W\!B_2$ string. In this paper, we construct a generalised BRST
operator that can be associated with a non-critical $W_{2,4}$ string, in which
$W\!B_2$ matter is coupled to the $W\!B_2$ gravity of the critical case. We
also obtain the complete cohomology of the critical $W_{2,4}$ BRST operator,
and investigate the physical spectra of the $s=5$ and $s=6$ string theories.
