How do I perform wudu with a cast or bandage on my hand?

When a cast, splint, or bandage covers part of a limb and removing it would cause harm, pain, or delay healing, the ruling is to perform masah (a single wet wipe) over the covering instead of washing the limb underneath, and to wash normally any part of the limb that remains exposed. One wipe with a wet hand over the majority of the covered area is generally sufficient, and — unlike wiping over leather socks (khuffain), which have a fixed time limit — masah over a medical bandage or cast has no set time limit; it can be repeated at each wudu for as long as the covering medically needs to stay on.

There's a real difference worth knowing between the Hanafi and Shafi'i schools here: the Hanafi position treats masah over the bandage/cast as sufficient purification on its own, without anything further required. The Shafi'i school takes a stricter view in certain injury scenarios — particularly when the bandage was applied over a wound in a state of impurity (i.e., not in a state of prior wudu) or covers more of the limb than medically necessary — requiring the person to also perform tayammum in addition to the masah, to combine both forms of purification (jam' bayna al-masah wa't-tayammum) out of caution. For most everyday injuries (a broken finger, a stitched cut, a sprained wrist in a brace), simply wiping over the covering and washing any exposed skin is what the majority of scholars, including mainstream Hanafi guidance commonly followed in South Asia, consider sufficient.

Source: general Sunni fiqh on masah 'ala al-jabirah (wiping over a splint/bandage); Hanafi position from Al-Hidayah and mainstream Hanafi fiqh texts; Shafi'i position requiring combined tayammum in certain cases, per Shafi'i fiqh manuals.