Throughout the year, and especially during this time of year as we approach Rosh Hashanah and Yom Kippur, we say many different types of tefillot in repentance. This list includes the daily tachnun, slichot, and the entire davening of Rosh Hashanah and Yom Kippur. Much of the text deals with the teshuva process, i.e. admitting that we have sinned, feeling sorry for it and making an effort not to try to continue in our ways, and begging the All Merciful God to forgive us and have mercy on us.
However, inevitably these tefillot also contain sections dealing with our oppressors, how they make our lives terrible, and asking God to destroy them. One short example from tachnun is: .עָשְׁשָׁה מִכַּעַס עֵינִי עָתְקָה בְּכָל צוֹרְרָי. סוּרוּ מִמֶּנִּי כָּל פֹּעֲלֵי אָוֶן כִּי שָׁמַע יְהֹוָה קוֹל בִּכְיִי - My eye is dimmed from anger, it has aged because of my adversaries. Turn away from me, all you workers of iniquity, for the lord has hearkened to the voice of my weeping. (Excerpted from here) This seems to deviate from the teshuvot tefillot where we identify a sin, try to repent and ask for forgiveness. Furthermore, though we would always prefer not to be punished, there is a realization in Judaism that punishment is not for nothing. It's objective is to goad us into the right path. So, however much we hate it, is it not instrumental in helping us clean us of our sins? Why do we ask for it be removed so desperately, without any caveat, like: Please remove these punishments God, if it would be in our best interest.