Yes, it’s definitely a reference to St. Paul. There’s been a lot written about how to interpret that verse. Here’s one good perspective.
Of course, God doesn’t need our sufferings any more than he needs our money. He’s complete and overflows with life. The universe was not made out of need, but out of gift. Our sufferings are therefore part of the gift he makes to us, weird as that sounds. As to “lack”, the term doesn’t refer to God lacking anything, nor to some inadequacy on Christ’s part but to our lack. For instance, God is the giver of all things and intends us to have all we need. Yet, in this world, people experience lack everyday. Why? In no small part because we (who are entrusted with the task of distributing what is needed for the common good) don’t supply the lack. So people starve or go thirsty. Is that because there is a lack on the part of the author of Being? No.
In the same way, Christ’s sacrifice is sufficient. But since he has made us participants in that sacrifice, our acts of sacrifice matter too. Had the apostles failed to proclaim the gospel, we would not have heard about it. We would lack. If somebody had not told me about Jesus, I would lack. If I don’t make the sacrifice of time to (for instance) reply to your note, you don’t get an answer to your question. You lack, not because Christ is insufficient, but because I don’t discharge my duty of trying to help. I don’t “offer my body a living sacrifice”, you therefore don’t receive the benefits of Christ’s sacrifice in the form of a letter from a brother in Christ who might have helped, but didn’t. You lack, not because of Jesus’ inadequacy, but because of mine.
I’m hoping that makes sense. The bottom line is: Christ mediates his grace to us through creatures, especially other people. We have nothing to offer God by ourselves. Even our ability to say yes is a gift of grace. But God has so willed that we can indeed make that offering—or not. When we do, it is joined to Christ’s offering and becomes part of the gift he makes of himself to the Church. The same is true of the mysterious sufferings of those who, seemingly, have nothing to offer but suffering. From a practical and utilitarian viewpoint, a bedridden victim of some disease appears to have “nothing to offer”. But then, so did Jesus when he hung on the cross and, to utilitarian eyes, accomplished nothing useful for six hours. In fact, of course, he accomplished the redemption of the world. A suffering soul likewise can join his or her sufferings to Jesus and, in his mysterious exchange of love, do great and wonderful things for others.