Edits

The trial has suggested that corruption in Mexico is as bad, if not worse, than many thought. 

“As bad …” needs another “as”; you can’t say “as bad … than many thought.” You could say “as bad as, if not worse than, many thought” — or, better and smoother, say “as bad as many thought, if not worse.” 

The filaments form visible spiral patterns within the rings, belying motions deep inside the planet that can be linked to its rotation speed.

“Belie” means to give a false idea of, or to show to be false. You should use something like “reveal.”

Courts can stop plaintiffs from shopping for judges in politically-charged politically charged cases, a new article argues.

No hyphen is called for in a compound modifier with an “-ly” adverb. Instead, say “politically charged cases."

Last year, the authority’s chairman, Joseph J. Lhota, said he would consider banning food, like as Washington’s Metro system does.

Avoid this use of “like” as a conjunction introducing a full clause. Make it “as Washington’s Metro system does.”

For the last two months, a team of three to five people have has laboriously swabbed away some of the varnish to allow the colors to come through.

Make it “has” to agree with the singular subject “team.”