Is it permissible to repeat dhikr to achieve a specific goal, such as repeating "There is no god but You; glory be to You! Indeed, I was of the wrongdoers" and seeking forgiveness to relieve distress?

There is no harm in repeating dhikr (remembrance of Allah), supplication (du'a), and seeking forgiveness (istighfar). It has been established in the Sunnah (definition) that supplications can be repeated three times, and sometimes even more. The Prophet, peace and blessings be upon him, supplicated against Quraish three times, and he supplicated for the tribe of Ahmas five times. A servant of Allah sought refuge from the Fire seven times and asked for Paradise seven times, and he supplicated for the sick seven times.