Card Chess with Playing Cards?
This sounds like the game Banqi (also called Half Chess or Dark Chess) but where you replaced the playing tokens with playing cards, the board size is increased, and the number of each available "piece" is always four.  This game is instead played on half of a Xianqi board with the pieces used for the Xianqi game (In order of strength: 5 pawns, 2 cannons, 2 knights, 2 rooks, 2 guards, 2 elephants, and a king for each player)

There are examples of it on the google play store under the name "Chinese Dark Chess" such as this one by Xidea.

The brief overview of the rules are that on your turn you may either move a revealed piece of your color, or flip over an unrevealed piece.  The first piece that is flipped by the first player determines the players' colors.

When moving a piece it may only move to an orthogonally adjacent square that is unoccupied, or it may move into an orthogonally adjacent square occupied by an opponent's piece that is of the same "strength" or lower, thereby removing the opponent's piece in the process.  In the case of ties the attacker wins and is not destroyed in the process.

Depending on which regional variant you are playing, cannons may have an alternate attack style, instead attacking as they do in a normal game of Xianqi by jumping over another piece to attack the second piece in its path, regardless of strength.  Further, a pawn may be able to capture a king (similar to how spies can capture field marshals in stratego).

The winner of the game is determined when the opponent has no legal moves available remaining.