There is, of course, no such thing as "cultural Marxism". It's just a term made up to make gullible idiots scared of people with dark skinned and who speak scary-sounding languages.
The term comes from German dissidents uncomfortable with existing capitalist, fascist or communist systems. Many of these theorists believed that traditional theory could not adequately explain the turbulent and unexpected development of capitalist societies in the 20th century. Critical of both capitalism and Soviet socialism, their writings pointed to the possibility of an alternative path to social development. Note that they weren't fascist, either.
I think what they were trying to get at was "social democracy", where basic requirements such as utilities, communications, roads, schools, etc were out of the control of capitalists, but not also handed off to communistic 5-year plans and such. Sort of betwixt and between capitalism and communism, but they hadn't developed the details of an ideology.