what does "one of several near misses for Godfrey?"
Mac had reacted quickly. The flight to Washington, he realized, would have to be postponed. He had headed for Swindal, and as he darted forward he must have passed McHugh—one of several near misses for Godfrey, which can only be understood in the light of the turmoil in the staff area.
"Cut it off!" Mac had told Swindal. He offered no explanation, but he was a member of the Presidential staff. The pilot had obediently reached for his instruments.

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A 'near miss' is when you almost hit something. It is often used in the context of aircraft and vehicle accidents. I assume that here, it meant that he passed close by him, by didn't actually see him. 'One of several' means that this situation had happened other times as well (with other people)
There's a bit of context missing here though - it's not clear to me who 'Godfrey' is.