My wife & me are Singaporean, we plan to visit relative for 7 days in Shanghai.

Nancy, if you can limit your stay in Shanghai to 6 days and fly from Singapore to Shanghai then to a third country or region, for example, Hong Kong, your children can make use of the 144-hour free transit policy so that the entry permit is not necessary. By the way, the free stay period starts to count from the midnight following your arrival, so actually you can stay for more than 144 hours.