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Nov 8, 2016 at 20:37 comment added Akhil Mathew Thank you for these references, @DylanWilson, and @TylerLawson! These sound exactly like what I was looking for.
Nov 7, 2016 at 17:36 comment added Tyler Lawson When $n$ is even I seem to recall that the elements $\theta^{(j)}(px)$ form a regular sequence in the free $\theta$-algebra on $x$ (you can pick off a coefficient which is a unit times $(\theta^{(j-1)} x)^{p}$ for $j > 0$), and so I think the result is the (completed) quotient of $\Bbb Z_p[\theta^{(j)} x]$ by the ideal generated by these elements. I don't know how explicit this can get.
Nov 7, 2016 at 16:48 comment added Dylan Wilson McClure computes some information about this in the H-infinity book, if I recall correctly. I think he gets the mod p K-theory of an arbitrary extended power of anything, and I remember him being particularly interested in Moore spectra at some point to deal with K-theory mod p^n.
Nov 7, 2016 at 16:38 history asked Akhil Mathew CC BY-SA 3.0