Does semen discharge due to severe constipation necessitate ghusl? And is wudu sufficient for secretions resulting from work and walking if their type is unknown, with the re-performance of Fajr prayer as a precaution?

The fluid that exits upon relieving oneself is usually wady, which is a thick, white fluid that comes out with urine, or before it, or after it. Its ruling is the same as that of urine.

Whoever is certain that what exited from him was maniyy (sperm/seminal fluid), and he had experienced arousal but the maniyy did not exit at that moment, then it exited after the arousal subsided (whether when relieving himself or at other times), he is required to perform ghusl (ritual bath) according to the majority of scholars. A woman is considered to be in a state of janaba (ritual impurity requiring ghusl) if her maniyy was displaced with desire and then exited without desire, according to the Malikis, Shafi'is, Hanbalis, Abu Hanifa, and Muhammad.

As for other discharges, ghusl is not required for them; rather, ablution (wudu) is invalidated by their exit, and they are pure according to many scholars. Accordingly, istinja' (washing after relieving oneself) is not required for them, nor is changing clothes if they are stained by them, and prayer is valid if one performs wudu after their exit. She does not need to repeat ghusl unless she knows that what exited from her was maniyy.