Technically to push press you do not even have to clean the bar into position if you can walk out of a rack with it.
Basically they are just a standing military press that is preceeded by a dip and drive with the legs. You are cheating with the legs to increase the amount of weight you can use, and proper timing will drastically increase the amount of weight that can be lifted.
If your goal is to work on your pressing strength, clean the bar once then press. If your goal is to work the entire shoulder girdle, execute a clean from the hang followed by a press for each individual repetition.
This exercise was first developed to help athletes learn the transition from the press to the jerk, as mastering the dip and drive is essential.
More details of the dip and drive, for those who care:
By the numbers:
Note: Where there are three numbers separated by a slash, such as x/y/z, this would indicate difference for athletes 150/170/190cm tall.
First Phase: The start: This is identical to the completed position of a front squat. The shoulder and hip joints should form a line bisecting the middle of the foot. The feet are approximately the width of the hips, and the toes are turned slightly outward. The head is tilted slightly back, and the bar is racked firmly on the anterior deltoids, with the elbows being maintained as high as possible without restricting air supply.
First Phase:
Pelvis behind heels: 5/5.5/6 cm
Knee angle during first phase: 180 degrees
Second Phase: The Dip: During the dip, the knees should be the only joint that flexes. The hips should stay in line with the shoulders, and this will cause the CCOG to shift slightly forward. Some athletes will allow the hips to travel slightly backward to compensate for this, but they successfully maintain the bar over the middle of the foot. As a general rule, the faster this phase is executed, the better
Second Phase:
Knee angle at end of second phase: 114 – 132 degrees
Elapsed time of second phase: 0.28 seconds
Speed of ABS during second phase: 0.85/0.98/1.1 m/sec
Barbell shift toward the athlete: 2 cm
Third Phase: Braking: This is where the athlete begins to resist the downward momentum of the ABS. This phase should be executed as fast as possible as well. The purpose of this phase is to prepare to thrust the ABS upward in preparation for the execution of the jerk. This phase ends when downward movement ceases.
Third Phase:
Knee angle at end of third phase: 99 – 111 degrees
Bar descends during third phase: 15/17/19 cm
Elapsed time of third phase: 0.12 seconds
Fourth Phase: Thrust: The athlete will elevate the ABS through a violent, explosive straightening of the legs. Once again, the only movement should occur at the knee joint as the legs extend. The purpose of this phase is to impart velocity to the bar in preparation for the next phase. This phase ends when the legs have fully extended.
Fourth Phase:
Pause at bottom before thrusting: 0.01 – 0.04 seconds
Knee angle at end of fourth phase: 175 degrees
Elapsed time of fourth phase: 0.19 seconds
Barbell velocity at end of fourth phase: 1.45/1.62/1.8 m/sec
From this point on, the key is to time the extension of the arms so that the energy generated by the thrust from the legs is transferred to the barbell. On average, superior athletes will begin pressing the bar when it is 6% below the height it was fixed at during the first phase.
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I do cleans and presses all the time, and I was wondering if push presses are where you only clean it up one time and then end up just pressing the bar for reps. 
