That thing from Amazon won't do much if the room itself has a lot of reverb. Treatment of the walls and surfaces will yield the most difference. I'd recommend a Zoom H5 over an H4n. Reason being that the H5 doesn't take 2 minutes to boot up and is far less finicky about what brand of SD card you use.
I record so many voices every day, I may give up video editing. No, I'm not giving up video editing. But I record several voice talents pretty much daily. And do lots of voice clean-ups to video shot by my staff in a variety of acoustically horrible places.
I have a Reflexion Filter (the original's of these things) from years ago in my office. I only use it cause i have it. My office is quiet enough, and some decorative Auralex foam kills enough reverb as a side benefit. Those devices do work, if they're made well. Some are made with cheap, inefficient materials and don't really do much. Also, if you're in a room that's quiet enough already, they can over-deaden the sound and make it way too flat.
As for "lots of reverb" mine has done quite well in some surprising places, but I have a unit that cost an arm and a leg and is relatively large-ish. But, there is such a thing as too much and you'd not get around it.
At the station we also use Zaxcom recorders, and we can be in a bad reverb room and get surprisingly clean audio. Not perfect, but CrumplePop's Echo Remover can take out the little that's left. Secret to a lav is get it really close to the throat, and record low.
EQ is about the best and first thing to spicing up audio in post, especially voices. I find the Logic EQ presets for voices give a lot of variety, and I can always fine one that works, sometimes with very minor tweaking.
And if this client has one room to set up in pretty permanently, agreeing with Red, treat the room. Just don't hand moving blankets, they're super flammable, not what I'd want around electronics. But regular bedding blankets, quilts, if they can't afford actual acoustics foam tiles.