What is the ruling on speaking kindly to a child in a video posted on Facebook and viewed by foreign men, with only the speaker's voice being heard, and does this fall under the category of "softening the speech" that is prohibited?

The woman's voice is not 'awrah (nakedness) according to the correct opinion. However, the Shari'ah prohibited softening and alluring the voice when speaking with non-mahrams, due to the Almighty's saying: ﴿Then do not be soft in speech [to men], lest he in whose heart is disease should covet, but speak with appropriate speech.﴾ [Al-Ahzab:32].

Therefore, there is no harm in addressing a child gently, as long as there is no softening or alluring that incites fitnah (temptation/discord). But if the recorded voice incites fitnah due to its softness and alluring tone, then it is not permissible to record it, as it falls under the general prohibition of softening speech.