On the first figure, you can see schematic diagram of typical thermionic electron gun from Wiki.with filament voltage(1-2) electrons are emitted and with bias(1-3) - focused.
and here is filament voltage vs. emitted electrons number, for different bias voltage.
it has a typical exponential behavior, but as you can see, there is kind of bump, the false peak near 3.4 volts. In the text this peak is explaining: „The false peak near 3.4 volts caused by region of filament that reaches emission temperature before tip„. but its still unclear for me. question is - what is the reason of this peak of current?