Is it permissible to consistently perform the Ṣalāt al-Ḥājah (Prayer of Need) before sunset every Friday, specifically at this time, and not concerning the permissibility of the prayer itself or performing it during a prohibited time?

There is no harm in performing Salat al-Hajah (definition) (prayer of need), but dedicating it and persistently performing it at a specific time not prescribed by the Shari'ah leads to an additive innovation (bid'ah idafiyyah). This is the commitment to a specific act of worship at a specific time without legitimate Shari'ah evidence. Al-Shatibi exemplified this by stating that it is the commitment to specific acts of worship at specific times for which no designation exists in the Shari'ah. Furthermore, Salat al-Hajah is not performed during disliked times (awqat al-karahah), such as after the Asr prayer until sunset, because its cause is delayed.