  We report results from continued timing observations of PSR J0740+6620, a
high-mass, 2.8-ms radio pulsar in orbit with a likely ultra-cool white dwarf
companion. Our data set consists of combined pulse arrival-time measurements
made with the 100-m Green Bank Telescope and the Canadian Hydrogen Intensity
Mapping Experiment telescope. We explore the significance of timing-based
phenomena arising from general-relativistic dynamics and variations in pulse
dispersion. When using various statistical methods, we find that combining
$\sim 1.5$ years of additional, high-cadence timing data with previous
measurements confirms and improves upon previous estimates of relativistic
effects within the PSR J0740+6620 system, with the pulsar mass $m_{\rm p} =
2.08^{+0.07}_{-0.07}$ M$_\odot$ (68.3\% credibility) determined by the
relativistic Shapiro time delay. For the first time, we measure secular
variation in the orbital period and argue that this effect arises from apparent
acceleration due to significant transverse motion. After incorporating
contributions from Galactic differential rotation and off-plane acceleration in
the Galactic potential, we obtain a model-dependent distance of $d =
1.14^{+0.17}_{-0.15}$ kpc (68.3\% credibility). This improved distance confirms
the ultra-cool nature of the white dwarf companion determined from recent
optical observations. We discuss the prospects for future observations with
next-generation facilities, which will likely improve the precision on $m_{\rm
p}$ for J0740+6620 by an order of magnitude within the next few years.
