[{"_id":1,"MyArray":[{"Name":"X","MyId":"Y"},{"Name":"N","MyId":"M"}]}]

db.collection.aggregate([{$addFields:{"ArrayId":{"$reduce":{"input":"$MyArray","initialValue":"","in":{$concat:["$$value",{"$cond":{if:{$eq:["$$this.Name","X"]},then:"$$this.MyId",else:""}}]}}}}}])