There aren’t any traditional rote prayers for washing hands or taking a shower. That exists in Jewish tradition, but we don’t have that. Consider this:
“Rejoice always. Pray without ceasing. In all circumstances give thanks, for this is the will of God for you in Christ Jesus.” 1 Thes. 5:16-18
“Rejoice in the Lord always. I shall say it again: rejoice! Your kindness should be known to all. The Lord is near. Have no anxiety at all, but in everything, by prayer and petition, with thanksgiving, make your requests known to God. Then the peace of God that surpasses all understanding will guard your hearts and minds in Christ Jesus.” Phil. 4:4-7
When I was a runner, there was a point in the race at which I had been trained to habitually “check down” that I was relaxed and in good form, prior to the stretch. It was just something we did every time we came around to particular points of the track during practice, so that in a race it would be automatic.
Life is busy now, and even when it is not busy it is easy to get absorbed in our own little projects. It is hard to remember to pray always or to rejoice. If washing seems to you to be a time to pray, perhaps you are primed to make washing a habitual trigger for checking for anxiety and making a request to God to hand it over to him. Once that is done (or if you aren’t anxious) then give thanks and rejoice.