We propose generalised $mathcal{N}=1$ superconformal higher-spin (SCHS) gauge multiplets of depth $t$, $Upsilon_{alpha(n)dot{alpha}(m)}^{(t)}$, with $ngeq m geq 1$. At the component level, for $t>2$ they contain generalised conformal higher-spin (CHS) gauge fields with depths $t-1$, $t$ and $t+1$. The supermultiplets with $t=1$ and $t=2$ include both ordinary and generalised CHS gauge fields. Super-Weyl and gauge invariant actions describing the dynamics of $Upsilon_{alpha(n)dot{alpha}(m)}^{(t)}$ on conformally-flat superspace backgrounds are then derived. For the case $n=m=t=1$, corresponding to the maximal-depth conformal graviton supermultiplet, we extend this action to Bach-flat backgrounds. Models for superconformal non-gauge multiplets, which are expected to play an important role in the Bach-flat completions of the models for $Upsilon^{(t)}_{alpha(n)dot{alpha}(m)}$, are also provided. Finally we show that, on Bach-flat backgrounds, requiring gauge and Weyl invariance does not always determine a model for a CHS field uniquely.
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