The strange example makes me confused - with + adverb : "Red items need dealing with immediately after the process..."
How correct and common of this structure?
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The issue which makes it sound strange is not with + adverb; it's -ing + adverb. In this case, need behaves as a modal verb and should be followed by an infintive form.
To deal with (something) is a phrasal verb, and there's not really much you can do about the with. It's part of the verb and needs to be there.
Red items need to be dealt with immediately after the process.