All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
Yes, it is permissible for you to fast one day or more per week with the intention of supererogatory fasting. However, if you only fast one day, do not fast on Friday, Saturday or Sunday. Abu Hurayrah narrated that the Prophet said: “None of you should fast on Friday unless he fasts on it together with a day before it or a day after it.” [Al-Bukhari and Muslim]
Besides, Abu Hurayrah narrated that the Prophet said: “…and do not exclusively choose Friday as a day of fasting among other days unless it occurs on a day that you regularly fast.” [Muslim] For more benefit, please refer to Fatwa 87228.
The same prohibition is also reported about fasting on Saturday only, and many scholars are of the view that it is disliked to single out Sunday for fasting. However, no Islamic text is reported about the prohibition of singling out other than these days for fasting as discussed in Fatwa 88751.
On the other hand, if by your statement “…if my wish is fulfilled…” you mean that you made a vow to fast if your wish is fulfilled, then if you had made such a vow, it is a conditional vow which is disliked as we clarified in Fatwa 89891. Moreover, some scholars are of the view that this is even forbidden if a person believes that Allaah would fulfill his wish because of his vow.
The Fiqh Encyclopedia reads: “Al-Qurtubi from the Maaliki school of jurisprudence said: “The vow is forbidden if it is feared that the person [who makes such a vow] believes that the vow necessitates that an urgent need would be fulfilled, or that Allaah will fulfill that need because of the vow [that he made], so it is forbidden for a person who holds that belief to make such a vow…”
However, if you had made such a vow, you are obligated to fulfill it if the condition upon which you conditioned your vow is fulfilled.
As regards doing that act of worship as a way of being grateful to Allaah for a blessing that one is blessed with, then this is a permissible and desirable matter.
Allaah Knows best.