Flux control in transformer
If both windings have the same number of turns and are driven with the same voltage then both will contribute equal amounts of flux to the core i.e. they will each produce a flux that is half the flux produced when just the primary was driven. Two halves make a whole.

If the turns are different and the voltages are scaled proportionately then this will also mean both windings contribute the same.

Anything else means the likelihood of an excessive current being taken and that current will not be related to core flux so, if you are trying to control core flux this way, it won't happen.