I run my own homebrew world and don't use published adventures. My prep takes on two parts - "arc" and "details" for lack of better names. "Arc" is me planing all of the plots and stuff, and usually I only put serious time into it when we have significant advances in the campaign or a lot of travel, so that's nto weekly. This could be things like advancing ongoing plots (both from what the PCs are doign and what else is happenign in the world), pulling in character backstory hooks, coming up with adventure hooks (either to advance a major plot or unrelated), etc.
Weekly I do the "detail". Work out specifics for what they are doing (or may do), stat up encounters, work up NPCs and personalities, etc. All of the specifics to run a situation. That said because the players have a lot of freedom in what they do and I run a bunch of plots at once, some of it may not get used for a few sessions if at all, so there is a lot of carry over from week to week.
I have a "prep" document that I update every week. Right now it's got details of the major city they are in: locations, organizations, and major NPCs, info on the order of knights they are dealing with (notable members and the code of the knights), details on a side quest about wiping out an enclave of wererats in the city (or at least tthat's what the characters think it is) which is more standard adventure format, some top-of-mind notes for character backstory that can come up in this city (one PC is from here, another from this duchy), light details on three other plots that the PCs might get involved in the short term, details on allied NPCs that they are having do things (sage research and talkign to the xenophobic halflings) and some "random encounters" if they shadowwalk.
As you can see, much of the detail stuff I can carry over from week to week so my prep time for any particular session isn't particularly heavy, takes me an hour to an hour an a half a week. Big advances in plot take more.