Is the prayer valid if one makes a mistake or forgets something while reciting the Quran within it, then says "Astaghfirullah" (I seek forgiveness from Allah) three times and re-recites the surah or verse, or must one begin with a new surah?

If someone forgets something from their recitation during prayer or makes a mistake in it: if the mistake is in Al-Fatiha, it must be corrected because the prayer is not valid without it. However, if the mistake is in something other than Al-Fatiha, then the prayer is valid, because the recitation after Al-Fatiha is a Sunnah. It is permissible to move to another verse, or to omit the surah, or to bow.