Through high-precision radio timing observations, we show that five recycled pulsars in the direction of the Galactic Centre (GC) have anomalous spin period time derivative ($dot P$) measurements -- PSRs J1748$-$3009, J1753$-$2819, J1757$-$2745, and J1804$-$2858 show negative values of $dot P$ and PSR J1801$-$3210 is found to have an exceptionally small value of $dot P$. We attribute these observed $dot P$ measurements to acceleration of these pulsars along their lines-of-sight (LOSs) due to the Galactic gravitational field. Using models of the Galactic mass distribution and pulsar velocities, we constrain the distances to these pulsars, placing them on the far-side of the Galaxy, providing the first accurate distance measurements to pulsars located in this region and allowing us to consider the electron density along these LOSs. We find the new electron density model YMW16 to be more consistent with these observations than the previous model NE2001. The LOS dynamics further constrain the model-dependent intrinsic $dot P$ values for these pulsars and they are consistent with measurements for other known pulsars. In the future, the independent distance measurements to these and other pulsars near the GC would allow us to constrain the Galactic gravitational potential more accurately.
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