So I see typically a TWT has a saturation point (gain).
Will it transmit a pulse with greater power? I know magnetrons can.
Pulsing is forcing the device into the saturated region of the amplifier. Done to make the amplifier more efficient in sync periods on TV signals. In the sync period there is no video (Vision) signal, so the amplifier can be run in the non-linear region, but under normal video period is opposite is the case, were the video needs to be amplified in a highly linear region of the amplifier. It is a power saving measure.
Personally I have never seen it done on TWT's, but on high powered UHF Klystrons running 15KW upwards to 40KW were amplifier efficiency can be around 30% for a typical high power amplifier running in linear region for TV. Pulsing pushed that figure up by a few percent.