I do not believe you've read the text incorrectly, and can suggest 2 possibilities:
A greensman associated with the film (and perhaps theatre) industry -- somebody who was responsible for the plants, flowers, landscaping etc. used on set during filming.
A greensman (or greenkeeper) who maintains the greens at a golf source or other sporting facility (perhaps associated with a school)
You could explore the possibilities by (a) looking for matching candidates in the 1939 register; (b) researching the 1930s film industry in the UK to see if the location involved supports an involvement with filming; (c) looking for golf clubs, schools and sports grounds in the relevant area.