I am currently trying to track down the relevant paper or documentation that describes the NpH algorithm that has been used in lammps. I am aware of the algorithm for the equations of motion as described by Shinoda et. al. in Phys. Rev. B, 69, 134103, but I can’t find any clear documentation for the NpH ensemble algorithm. If someone could help point me in the right direction it would be much appreciated.

That is correct. The best source for the equations of motion of the
barostat variables is the citation given on the LAMMPS doc page:

Tuckerman, Alejandre, Lopez-Rendon, Jochim, and
Martyna, J Phys A: Math Gen, 39, 5629 (2006).

Also, Mark Tuckerman's book
"Statistical Mechanics...." devotes a whole chapter to isobaric ensembles.
There you will find a proof that the distribution of positions and momenta
generated by fix nph actually corresponds to the NpH ensemble. It should
be noted that the enthalpy that is conserved differs by O(1/N) from the
normal definition of enthalpy and is exactly equal to the quantity printed
by the thermo keyword etotal, when you use fix_modify energy yes.

Look at the output from this script (columns v_hlammps v_hset v_hnph) to
see how
this works in practice.