(EDIT: I thought you were asking about a diode in parallel with the body diode. My bad.)
The diode as you suggested would not "cancel the body diode" because it wouldn't be engaged to begin with: with MOSFET off and positive supply on its drain, body diode would be off.
With MOSFET on, the body diode would be shorted out and it would be off as well.
The only time the body diode is engaged or turned on is when there is a current flow in reverse direction through the MOSFET, meaning positive voltage on source electrode and negative voltage on drain for N-channel type MOSFET.
As "AnalogKid" has already pointed out, the only useful scenario for your diode is to block this reverse current, but in these situations often a diode drop/loss presents too much power loss or too much heat generation in the diode, and another MOSFET is added and turned in the opposite direction to prevent current in either direction when MOSFETs are off, and to have a significantly lower voltage drop and power loss when MOSFETs are on, as in the circuit below:
simulate this circuit – Schematic created using CircuitLab
(Thanks to the user "AnalogKid" who pointed out that MOSFETs in this type of arrangement have sources tied together, which allows using a common driver for both.)
(OLD ANSWER below, assuming a diode parallel to body diode:)
Yes, you can have it and it's frequently done.
Most of the time, the body diode is actually a positive addition which protects the MOSFET in some cases or helps its function in other case (for example, using it in synchronized rectifiers as a rectifier diode for a brief moment until MOSFET turns on and reduces the voltage drop).
However, it is usually slower than the ultrafast or Schottky diodes you can get, and it also has a higher forward voltage drop.
If your MOSFET is doing some very high speed switching, its body diode's typically long reverse recovery time can waste too much power at MOSFET turn-off as it would let a significant amount of current flow through it backwards, which not only wastes more energy and produces more heat because such current is multiplied with the larger Vds at that moment, but it also risks destroying the MOSFET.
When you add a faster diode with a lower voltage drop, the body diode is not even engaged.
There are MOSFETs whose body diode is intentionally made faster and having a lower voltage drop or a separate diode die is added/integrated with the MOSFET.