You should be able to piece this one together. Here are the functions you'll need:
OR(logical1, logical2...) and replace any number of logical references with the expressions you want evaluated.
AND(logical1, logical2...) and replace any number of logical references with the expressions you want evaluated.
ISPICKVAL(picklist_field, text_literal) and replace
picklist_field with the merge field name for the picklist; replace
text_literal with the picklist value in quotes.
text_literal cannot be a merge field or the result of a function.
- For a blank picklist, you want to use
"" for the
NOT(logical) and replace
logical with the expression that you want evaluated.
You should be able to compose the above into one working expression! I recommend you make an attempt, and you will learn more from that than from someone posting something for you to copy-paste.
In plain English your rule should read something like: Throw an error either when Type is X or Y and Customer Management is blank, or when Type is A or B and Customer Management is not blank.
Now that you've added your attempt to the question, I feel better about showing you how I would write it. I do think it might be better to split into separate validations. You'll give your end users clearer error messages. That said, here are the two separate validations you want to perform. If you want to use just one rule, you would do
Enforce Not Blank
ISPICKVAL(Type, "Client - Group Contract")