The phrase emphasizes a stressful or anxious condition that prevents sleep. Before the job interview, I couldn't get a wink of sleep.
Pro Tip 2/3
Not Literal
It may be an exaggeration, not always meaning zero sleep, but very little or poor quality. With the baby crying all night, we didn't get a wink of sleep.
Pro Tip 3/3
Cultural Expressions
It's common in English to use 'a wink' as a way to talk about sleep, though other cultures may not do this. At the noisy hotel, I didn't get a wink of sleep.