I had the exact same thought 
39 across seems to be カブ (at least if you google 株 根菜 you get lots of images of radishes and beets and carrots and the like)
38 down might be キカイ？
For 34 across I’m thinking things like 出勤 but of course it is too long… the only word that ends with 勤 and that fits would be ヤキン (夜勤) but I don’t see how this fits the prompt…
Bleh, this is how the thread keeps going idle - someone posts, and I put off replying back, and then everyone forgets about it…
20-down ヨシ (because アシ and ヨシ are both readings for 葦, though I’m missing why you’d want to avoid the アシ reading, exactly)
14-across… ツヨサ? ツヨミ? …ツヨキ? Really gotta get that big word down the middle solved…
Why not シキン? As in 資金 
Then 33-down イシ
25-across シカイ
27-down ロカ
16-across… モノクロ?
Then 23-down could be, uh… スイミンガクシュウ