Less oxygen to working muscles (in this case cardiac muscles that pump the heart) means less oxygen being used to be converted to water in the electron transport chain. Therefore less ATP production meaning less cardiac output function. I believe this is correct. Someone check my work.
It doesn't initially. At first there will be a strong sympathetic system response which increases cardiac output in order to maintain oxygen delivery to tissues in the face of hypoxia. If the hypoxia goes uncorrected then, yes, hypoxic heart muscle doesn't contract as effectively and cardiac output starts to fall.