How does one perform tayammum when there is a plaster on the hand due to a bleeding skin allergy that is affected by water and dust?

If removing the adhesive and performing tayammum (definition) are possible without harm, then the adhesive must be removed, tayammum performed, and then the adhesive reapplied. However, if removing the adhesive or the contact of dust with the skin would cause harm, then you should pray in your current state without wiping over the adhesive during tayammum. Your prayer will be valid, and you will not be required to repeat it. In this case, you are considered to be in the ruling of "one who has lost both means of purification" (faqid al-tahurayn), and both ablution (wudu') and tayammum are waived for you. You pray in your current state, for "Allah does not burden a soul beyond its capacity."