Does semen transfer from clothes it has touched to other objects, making them impure, which affects the validity of prayer if these objects are used?

Semen is pure and not impure, according to the preponderant opinion of scholars. Therefore, items touched by your hand with semen on it do not become impure, nor do objects that subsequently touch those items. Even if its impurity were asserted, the transference of impurity has specific conditions.

Semen necessitates ghusl (ritual bath) by consensus, based on the Prophet Muhammad’s (peace be upon him) saying, "Water is from water." If you did not perform ghusl after seeing semen that exited you during sleep, you must perform ghusl and make up all the prayers you performed while in a state of janabah (major ritual impurity).