Is it permissible to recite the Quran or dedicate a specific wird (regular portion of devotion) with the intention of facilitating a certain matter, such as agreeing to recite parts of the Quran to complete it three times with the intention of bringing relief to Muslims, or dedicating a specific dhikr (remembrance of Allah) like "There is no power and no strength except with Allah" a certain number of times with the intention of seeking relief?

There is no harm in reciting the Qur’an or engaging in remembrance (dhikr) with the intention of attaining good or averting evil, as this is a means of seeking nearness to Allah through righteous deeds. There is also no harm in advising one another to perform such acts to achieve that purpose, provided that one does not believe in a specific effect for a particular act of worship in achieving the desired outcome, and provided that people do not gather in a specific manner that makes the unsanctioned resemble the sanctioned. Otherwise, that would be an additional innovation (bid'ah). Indeed, making the unsanctioned resemble the sanctioned is a reprehensible innovation.