HP Scanjet G4050 Connecting

Every Apple operating system since Snow Leopard (10.6) has been 64-bit, and Mojave is no exception. Unlike Catalina, Mojave can still run 32-bit and 64-bit applications, and VueScan will work just fine on Mojave (or Catalina).

The last update (HP Easy Start) for this G4050 Scanner was in June 2019, and that was for High Sierra (10.13.6), not Mojave, which HP shows no support for at all.

This scanner model is supported without any HP software/drivers by the paid, 64-bit VueScan product.

Mojave is not 32-bit, it is a 64-bit operating system. You can use the current, 64-bit version of VueScan on Mojave. The trial version may not give you full access to the application's features on your particular scanner.

I have owned VueScan Professional since 2010, and still use it with a Canon scanner that was discontinued before then. I have never paid an upgrade fee for the Professional version, and receive any form of compensation for recommending the product here.

One of the issues in describing 32-bit and 64-bit is the introduction of additional, and confusing vocabulary. You can read more about 32-bit and 64-bit, if you want. All modern processors used in PC and Macs are now 64-bit. Depending on the developer, legacy applications could be written so as to be 32-bit compatible, or 64-bit compatible. Nothing you have to be concerned with as long as you are running Mojave or earlier.

On Catalina, any 32-bit coded application will be unsupported, and the individual developers, if they choose to ship product for Catalina, will need to revise (if not rewrite) their 32-bit applications to become 64-bit applications. It is the developer's responsibility to make this happen, not Apple.

Errata:

and receive any form of compensation for recommending the product here.

Should have read:

and do not receive any form of compensation for recommending the product here.

You are welcome. Bonus that VueScan supports your Brother device too.