Learn Finnish: Vedimme autoa kaikin voimin. - We pulled the car with all our might.

This is the instructive case, which is almost always used in the plural and ends in -n. It expresses the instrument, means, or manner by which an action is done. In this fixed phrase, both the modifier kaikki (all) and the noun voima (strength) take the instructive plural ending -in to mean "by means of all strength".

Why does the word kaikki become kaikin with a single k instead of "kaikkin"?

The instructive case is formed using the weak plural stem of a word. The plural stem of kaikki is kaikki-. Because the instructive ending -n closes the syllable, it triggers consonant gradation, weakening the strong double kk to a single k. This gives us the form kaikin.

The English translation says "with all our might". Why is there no possessive suffix in the Finnish sentence?

While you could technically express this using the adessive case with a possessive suffix (kaikilla voimillamme), fixed idiomatic phrases in the instructive case usually omit possessive suffixes. The ownership of the "might" is already obvious from the verb vedimme (we pulled).

Why is the object autoa in the partitive case rather than taking a total object ending?

The verb vetää (to pull) typically takes a partitive object when the action is an ongoing process (an irresultative action). The sentence describes the continuous effort of pulling the car, rather than the completed action of having successfully pulled it to a final destination.