We report on the discovery of four millisecond pulsars (MSPs) in the High Time Resolution Universe (HTRU) pulsar survey being conducted at the Parkes 64-m radio telescope. All four MSPs are in binary systems and are likely to have white dwarf companions. In addition, we present updated timing solutions for 12 previously published HTRU MSPs, revealing new observational parameters such as five proper motion measurements and significant temporal dispersion measure variations in PSR J1017-7156. We discuss the case of PSR J1801-3210, which shows no significant period derivative after four years of timing data. Our best-fit solution shows a period derivative of the order of $10^{-23}$, an extremely small number compared to that of a typical MSP. However, it is likely that the pulsar lies beyond the Galactic Centre, and an unremarkable intrinsic period derivative is reduced to close to zero by the Galactic potential acceleration. Furthermore, we highlight the potential to employ PSR J1801-3210 in the strong equivalence principle test due to its wide and circular orbit. In a broader comparison with the known MSP population, we suggest a correlation between higher mass functions and the presence of eclipses in `very low-mass binary pulsars, implying that eclipses are observed in systems with high orbital inclinations. We also suggest that the distribution of the total mass of binary systems is inversely-related to the Galactic height distribution. Finally, we report on the first detection of PSRs J1543-5149 and J1811-2404 as gamma-ray pulsars.
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