Analytical investigations are performed for pressure driven flow of an electrically conducting, incompressible and viscous fluid in a polyelectrolyte-grafted nanotube by using Bessel functions. Nanofluidic tubes whose walls are covered by polyelectrolyte materials, named the fixed charge layer (FCL), are identified as soft nanotubes. The flow relies on an externally imposed pressure gradient and an induced reverse electroosmotic force produced by the streaming potential field which is spontaneously developed due to the ionic charge migration with the fluid flow. Many parametrical ranges are determined to ensure the validity of Debye-Hckel approximation. The analysis is based on the solutions of the linearized Poissson-Boltzmann equation and modified Navier-Stokes equation. To obtain the streaming potential, we use a numerical treatment to solve an integral equation governing the streaming potential. Finally, the electrokinetic energy conversion efficiency is studied. The result shows that both the streaming potential and energy conversion efficiency monotonically increase with the FCL thickness d increasing. However, they present a monotonic decrease trend with the increase of K, which is the ratio of the characteristic scale of the mobile charges to the fixed charge within the FCL. We compare the results in a soft nanotube with those in a rigid one, whose zeta potential is equal to the electrostatic potential at the solid-polyelectrolyte interface of the soft nanotube. We find that the electric potential in a soft nanotube is higher than that in the corresponding rigid nanotube, which results in a larger streaming potential in the soft nanotue. Moreover, for the parameter ranges considered in this work, our results show that the electrokinetic energy conversion efficiency in a soft nanotube is 1.5-3 times higher than that in a rigid nanotube. These findings are important for investigating the streaming potential and electrokinetic energy conversion efficiency in soft nanotubes. They can be used as a kind of new method to enhance the energy conversion efficiency of the electrokinetic transport in nanotube.