Are our hardships in this life erasing punishments we might receive in Jahannam?

Yes, we know from the hadith in Sahih Bukhari, the Prophet ﷺ said that every difficulty a Muslim goes through, even if it's a prick of a thorn on your skin, that tiny amount of pain is an expiation for sins, for mistakes. So always have good thoughts about Allah ﷻ. Sometimes Allah will even put a difficulty in your life to erase your sins, to raise you to a rank that maybe you would have not been able to achieve without that difficulty. So always think of it in that way. And remember the verse, that with one difficulty, many doors of ease will open for you, inshallah.