From your description, it sounds like you are driving into Manhattan from either lower Brooklyn or Queens, using 1 of the 3 East River bridges -passing thru Chinatown on your way to the Holland Tunnel, with Canal Street as the obvious major east-west cross town route.
On any given day, weekends or otherwise during the work week - I avoid using Canal Street unless it's early morning like before 7 AM or after 8 or 9 PM at night, when traffic eased up and blocking the box usually isn't a problem.
Otherwsie, including Saturday afternoon or evening return, avoid using Canal Street or even Broome Street to approach the HT, go around it using Church Street to 6th Avenue (from City Hall area, easy from the Brooklyn Bridge) and left turn on Canal Street, then keep to the right - 2 traffic lights and you are at the entrance to the HT. This only works on weekend & off-peak on weekdays as there are restrictions otherwise except for buses, etc.
Another alternate would be Varick Street north to Canal, keep right for the entrance to HT but it could sometimes jam up on weekends.
Aside from GPS/Google Nav live via smartphone, good old car radio tuned to news 880 or 1010 news are your friend, for live & up-to-the-minute traffic reports every 10 minutes.
Just leave a bit early to allow for extra travel time, ignore those that honk behind you, and you will be like the 90% that don't block the box. Our friends from Canada, California & overseas will NOT drive in & around NYC for these & other reasons.
Another annoying fact behind the wheels is that NYC is NO right turn on red, and out-of-state drivers often ignore or pretend to not know, with the worst offenders usually ignoring all traffic rules completely - speed thru local streets, don't yield to people crossing legally, etc. etc. - if only NYPD and its traffic agents would enforce these laws and ... Blocking The Box tend to get more attention around the fall holiday season, especially on "Gridlock Alert Days"
Now that we've managed to scare the daylight out of you for not driving around NYC, you should be fine on a Saturday - unless the POTUS is coming to town, bringing AFO and company with him ....
