Drink problem has become prevalent in the UK over the past 30-40 years, but grammatically only the gerund drinking can be used as an adjective (type of problem) or even if reworded, a noun for an activity (problem with drinking). For anyone outside the UK “drink problem” would refer to a situation in which something is wrong with your drink (say a fly in your Coke or whiskey in your vodka tonic).
@Zhang
In this case delicate means something that could likely make the person uncomfortable or embarrased.