I’m no expert on flow variables but I think you would need a flow.get function before your media player node to pass the message in and possibly change your flow.set with:
Flow.Set Function
var var_k_play = "light.study_light";
flow.set("var_k_play",var_k_play);
msg.var_k_play=var_k_play;
return msg;
Flow. Get Function
msg.var_k_play=flow.get('var_k_play');
return msg;
Then change your Entity_ID in the media node to
{{var_k_play}}
as this is referencing msg.var_k_play and not the variable you defined in the function -I think
So something like this:-
[{"id":"ccc8b9d6.541e58","type":"function","z":"3f648309.933bc4","name":"Flow.get","func":"msg.var_k_play=flow.get('var_k_play');\nreturn msg;","outputs":1,"noerr":0,"initialize":"","finalize":"","x":260,"y":720,"wires":[["3da8b512.e8e1b2"]]},{"id":"57339543.09db04","type":"debug","z":"3f648309.933bc4","name":"function get","active":true,"tosidebar":true,"console":false,"tostatus":false,"complete":"payload","targetType":"msg","statusVal":"","statusType":"auto","x":630,"y":720,"wires":[]},{"id":"3da8b512.e8e1b2","type":"api-call-service","z":"3f648309.933bc4","name":"","server":"57ce5691.fd6e48","version":1,"debugenabled":false,"service_domain":"light","service":"turn_on","entityId":"{{var_k_play}}","data":"","dataType":"json","mergecontext":"","output_location":"","output_location_type":"none","mustacheAltTags":false,"x":450,"y":720,"wires":[["57339543.09db04"]]},{"id":"f570eb8b.5ea69","type":"inject","z":"3f648309.933bc4","name":"","props":[{"p":"payload"},{"p":"topic","vt":"str"}],"repeat":"","crontab":"","once":false,"onceDelay":0.1,"topic":"","payload":"","payloadType":"date","x":120,"y":720,"wires":[["ccc8b9d6.541e58"]]},{"id":"83ad44f6.9f769","type":"function","z":"3f648309.933bc4","name":"Flow.set","func":"var var_k_play = \"light.study_light\";\nflow.set(\"var_k_play\",var_k_play);\nmsg.var_k_play=var_k_play;\nreturn msg;","outputs":1,"noerr":0,"initialize":"","finalize":"","x":260,"y":660,"wires":[["dbfca78.1985ed8"]]},{"id":"1966a243.5fca16","type":"debug","z":"3f648309.933bc4","name":"function set","active":true,"tosidebar":true,"console":false,"tostatus":false,"complete":"payload","targetType":"msg","statusVal":"","statusType":"auto","x":630,"y":660,"wires":[]},{"id":"dbfca78.1985ed8","type":"api-call-service","z":"3f648309.933bc4","name":"","server":"57ce5691.fd6e48","version":1,"debugenabled":false,"service_domain":"light","service":"turn_off","entityId":"{{var_k_play}}","data":"","dataType":"json","mergecontext":"","output_location":"","output_location_type":"none","mustacheAltTags":false,"x":430,"y":660,"wires":[["1966a243.5fca16"]]},{"id":"1573a07c.d01dc","type":"inject","z":"3f648309.933bc4","name":"","props":[{"p":"payload"},{"p":"topic","vt":"str"}],"repeat":"","crontab":"","once":false,"onceDelay":0.1,"topic":"","payload":"","payloadType":"date","x":120,"y":660,"wires":[["83ad44f6.9f769"]]},{"id":"57ce5691.fd6e48","type":"server","name":"Home Assistant","addon":true}]