Database
[{"_id":1,"MyArray":[{"Name":"X","MyId":"Y"},{"Name":"N","MyId":"M"},{"Name":"X","MyId":"Y"}]}]
Query
db.collection.aggregate([{$addFields:{"ArrayId":{"$reduce":{"input":"$MyArray","initialValue":"","in":{$concat:["$$value",{"$cond":{if:{$and:[{$eq:["$$this.Name","X"]},{$eq:[{$strLenCP:"$$value"},0]}]},then:"$$this.MyId",else:""}}]}}}}}])