What is the ruling on delaying the Dhuhr prayer until one reaches home to perform it with full purity, and what is the last time for performing the Dhuhr prayer?

The last time for Dhuhr prayer is when the shadow of a thing becomes equal to its height, which is the beginning of the Asr prayer time, based on the Prophet's (peace be upon him) saying: "The time for Dhuhr is when the sun declines (from its meridian) and the shadow of a man is like his height, as long as the time for Asr has not yet arrived."

If you return home before the time for Asr enters, there is no harm in delaying the Dhuhr prayer until you are home. However, if you return after the time for Asr has entered, then you must perform the prayer at its prescribed time.

Regarding the impurity that affects clothing, if it is vaginal discharge, the preponderant opinion is that it is pure, but it nullifies ablution (wudu'). If it is an impurity like urine or menstrual blood (istihada), then the clothes must be purified and ablution performed before prayer. If you suffer from incontinence (sals), then you must take precautions and perform ablution after the time for prayer has entered, then pray.