(おぎなう) To supplement – like 補充, but the nuance is that you are filling a vacancy. For instance, nutritional supplement. Substitute teacher. Assist in the completion of something (I helped write the speech). To make up for (i.e. Mitsubishi bank loaned money to Toyota to make up for their shortfall =みつびし銀行はトヨタの赤字補いました。)     $$$TTT
(ほきょ) To reinforce.
する(ほじゅうする) To supplement, (a shortfall or deficit) or replenish (fluids, vitamins to replenish strength, restock the store-room) it’s like 補う, but the nuance is that you are restocking a supply of something which has dwindled.    PPPTTT
(ほい) Unlike 補充 or 補う、補遺 is used when you have given someone all they asked for, and you supplement that with a little extra something, just to be nice. A Sunday newspaper (remember those?) with the 24-page color supplement. A museum ticket that comes with a supplementary pamphlet explaining about the exhibits. Close to the word おまけ - a freebie.