Lede - There are a few issues that need fixing here.
• ... was one of the worst tropical cyclones to ever impact the South Pacific kingdom of Tonga. - Worst, how? This could imply several things. Was it deadly or damaging? In this case, it'd be better to go with something more specific such as "costliest", "devastating" or "catastrophic".
• Forming out of a near-equatorial trough in mid-December 2001, the precursor to Waka initially struggled to gain strength - Although I understand what you mean, this part could be misleading. Did the precursor to Waka form out of the trough, or was the trough the precursor to Waka?
• The storm gradually was able... - Doesn't read all too well.
• ... and 104.2 million paʻanga ($51.3 million USD) was wrought in damage. - Technically nothing wrong here, but consider using the currency sign (T$) instead.
• Hundreds of structures were destroyed and much of the nation's agriculture was destroyed. - A bit repetitive; could be altered a bit.
Overall, the prose is a little verbose and clunky; try to smoothen it out a bit.
Meteorological history - Pretty solid; no more than a handful of minor issues.
• You link sea surface temperature twice, though under different terms (initially "warm waters" and later on the actual term "sea surface temperatures"), which could be a bit confusing. I'd suggest using "warm sea surface temperatures" or something along those lines instead, and de-linking the second one.
• Later that day, the cyclone attained its peak intensity as a Category 4 severe tropical cyclone with winds of 185 km/h (115 km/h) and a barometric pressure of 930 mbar (hPa; 27.46 inHg).[2] The JTWC assessed the storm to have attained similar one-minute sustained winds upon peaking. - I'm guessing the first assessment was done by the Fiji Meteorological Service, and that the winds measured by the agency were one-minuted sustained as well? Perhaps you could clarify a bit.
• Gradual weakening took place over the following few days and the system retained gale-force winds through January 5; however, its pressure continued to decrease. - This sentence is sort of bloated and doesn't read too well.
• I fixed some minor issues, but it could still use a copy-edit.
Preparations and impact - Same as MH, basically.
• Many residents on the small island of Niuafo'ou, about 35 km2 (13.5 mi2) in size, evacuated to other islands prior to Waka's arrival. - Why do you mention its size? Did it have any effect on local preparations or impact?
• According to local reports, nearly every tree in Neiafu had been downed by high winds and "It is like the island [was] sandblasted." - The inline quotation is kind of terse and loose.
• ... about 200 homes in the city were severely damaged or desroyed. - Typo. Also, it'd be better if you switched around "severely damaged" and "destroyed".
You mention Waka passed directly over Vava'u in both the lede and the Impact and preps section; however, it isn't mentioned in the MH, which is supposed to cover any major aspects regarding its storm path.
It's looking much better now. A few problems left I see...
• The storm attained its peak intensity as a Category 4 severe tropical cyclone on December 31 with winds of 185 km/h (115 mph); it passed directly over Vava'u at this strength. - Kind of an awkward structure here, especially for a lede sentence.
• ... the system entered a region which favored - Clause should be restrictive.
