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With a single eye to something vs With one eye on something

with a single eye to somethingexclusively concerned with or concentrating on smth.:

  • His predecessors are the only Roman emperors who can be said to have ruled with a single eye to the welfare of their subjects.

with one eye on something —(also: with half an eye on something)

1. directing one’s attention only partly to smth. (while preoccupied with smth. else):

  • She sat at the head of the table stirring her tea, with one eye on baby Paul, who was fretful in his little chair.

2. simultaneously thinking about or preoccupied with smth.:

  • He still told jokes, but they were the jokes of a man who couldn’t see anything funny any more, the jokes of a man with one eye on the door.

Note: Neither expression correlates in meaning with the phrase with one eye closed(also: with one eye shut) do smth. easily or with little effort:

  • You might cope for a while with one eye closed … but you wouldn’t keep it up for long—nor would you perform at maximum ability.