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Regarding the ayah:

وَإِذْ قِيلَ لَهُمُ اسْكُنُوا هَٰذِهِ الْقَرْيَةَ وَكُلُوا مِنْهَا حَيْثُ شِئْتُمْ وَقُولُوا حِطَّةٌ وَادْخُلُوا الْبَابَ سُجَّدًا نَّغْفِرْ لَكُمْ خَطِيئَاتِكُمْ سَنَزِيدُ الْمُحْسِنِينَ
_{Qur'an 7:161}

Some translations (sourced from Islam Awakened) are:

And remember it was said to them: "Dwell in this town and eat therein as ye wish, but say the word of humility and enter the gate in a posture of humility: We shall forgive you your faults; We shall increase (the portion of) those who do good."
_{-- Yusuf Ali (Saudi Rev. 1985)}

And [mention, O Muhammad], when it was said to them, "Dwell in this city and eat from it wherever you will and say, 'Relieve us of our burdens,' and enter the gate bowing humbly; We will [then] forgive you your sins. We will increase the doers of good [in goodness and reward]."
_{-- Sahih International}

And (remember) when it was said to them: “Reside in this town and eat from there whatever you wish, and say: ‘(O Allah) forgives our sins,’ and enter the gate prostrating (bowing with humility). We shall forgive you your sins. We shall increase (reward for) the righteous people.”
_{-- Abdul Hye}

I'm a bit puzzled as to why there's multiple, somewhat different translations of the part highlighted in bold.

Question: In Qur'an 7:161, why is "say the word of humility" translated in various ways?

The main relevant Arabic word seems to be حطة (ḥiṭṭa) = "alleviate". I'm not sure how this translates.

The reason for the multiple translations is that there is a difference of opinion regarding the meaning. Ibn Abbas mentions two interpretations of the meaning of the word "حِطَّةٌ" in 2:58 (which is a similar verse to 7:161):

[It means] say: "forgive our sins" ; as it is also said that this means: say: "there is no god save Allah" (Ibn Abbas)

Here, he mentions another interpretation that you haven't come across: that it may also mean "there is no god save Allah."

This difference in opinion is primarily because this is one word that we are trying to find the meaning of expressed in a full sentence. In the English language it would be similar to an exclamation "Lower!". It could mean lower one's sins (i.e. forgive us) and it could also mean lower oneself (i.e. humbling oneself). It is not clear to interpret which one it is with just the language.