What is the meaning of 左右着 here? How would you translate 左右着他们 here? What other phrase could I substitute for 左右着他们 here?

在古代的时候，科学水平还没那么高，人们对自然的认识也是有限的。因此，碰见灾难的时候，就会认为后神灵在左右着他们，所以人们祈求于神灵。

就会认为后神灵在左右着他们，
then they might believe the Gods/spirits are ??? 

左右着他们
stopping them
punishing them
near them
ruling them
guiding them
around them
佐 means 'assist' ; 佑 means 'protect'. &quot;佐佑&quot;(to assist and protect) is different from &quot;左右&quot; (to make someone go left or right = &quot;to influence&quot;)

左右 http://www.cantonese.sheik.co.uk/dictionary/words/332/
[v] influence; master; control

In your sentence, 左右着他们 would mean: &quot;influence them&quot; or &quot;control them&quot;
More example:
左右朝政 (exert influence/control over the government's policy; influence the government)
大企業以財力左右政局 (Large enterprises use financial resources to influence/ control politics)
左右局勢 (exert influence/control over the situation; influence/control the situation)
憑他一人之力是無法左右大局的 (With his power alone, he can’t influence/ control the big picture)
左右(to influence) might or might not had been evolved from 佐佑(to assist and protect). In any case, its meaning in modern Chinese is completely different from 佐佑
左右 = to make go right or left = to influence; to control
佐佑= 佐助 (to assist) 庇佑(protect)