How did Efraim cut out Henry?

I don't think there are too many intricacies to it. Efraim learned about Henry's deal in the first place because their driver Bashkim had contacts to the Albanian ministry of defense (through his cousin, if I remember correctly). So the assumption would be that Efraim simply used these very same contacts to approach the Albanian government about a better deal to buy the rounds directly from them (he would only slightly have to overbid Henri's price and would still ultimately make a much better deal).

The reason they had Henry in it was because he had the contacts to make the deal happen in the first place and now that Efraim found another way, he's trying to take things into his own hands. It's shown repeatedly how surprisingly simple and direct these supposedly huge weapon deals are made, so it wouldn't be much of a stretch to all this coming down to a few phone calls between Efraim and whoever in the Albanian government.

As to the question of payment. It's not unlikely that they hadn't paid Henri yet, or certainly not fully. While it's quite an important deal, Henri isn't someone you...don't want to pay anyway, so this might very well work based on trust to some degree. It seems quite clear that they themselves aren't paid until the US military has their goods (as we have also seen with the Beretta deal). They might get paid for individual deliveries, but they likely aren't paid entirely in advance. And we're talking about huge deliveries for multiple millions of dollars here, AEY might very well not have had that amount of money as backing. So the way of the money would likely be from the US military, to AEY, (to Henry,) to Albania, minor advance payments for trust building notwithstanding.

But...I'm not an arms dealer either, nor an expert on any other kind of semi-legal international large capacity trades.

_{I do, however, agree with the commenters that Efraim merely tried to cut out Henry and it didn't actually work, be that because the Albanian government was more loyal than Efraim thought or because someone else warned Henry. We also see at the end that Henry made money in the deal. But that's marginal to the question.}