Streamed or ran away with a hearty cry! (6)
I believe the answer is:
flowed
'streamed' is the definition.
(I've seen this in another clue)
'ran away with a hearty cry' is the wordplay.
'ran away' becomes 'fled' (I've seen this before).
'with' means one lot of letters goes inside another.
'a hearty cry' becomes 'ow' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'fled' enclosing 'ow' is 'FLOWED'.
'or' is the link.
Can you help me to learn more?
(Other definitions for flowed that I've seen before include "What the champagne did" , "Moved steadily in a stream" .)