You asked me whether UUs pray. They do, and they have lovely prayers which they say in unison as part of their church services.
The deeper considerations to your question are probably to whom and for what do UUs pray. The answer to the first question is, it varies depending on the person doing the praying. What/who does he/she consider his/her higher power to be and from what/whom does he/she draw his/her inspiration and solace? The best prayers are prayers of forgiveness, gratitude, and the request for direction. My prayers are pretty simple: Dear God help me to follow your will not my will - Thy will be done, and Thank you for the three good things that I have been blessed with today."
You might consider the suggestion of Dr. Ira Byrock who says that when we encounter people who are dying (and we all are dying all the time whether we are aware of this or not) it helps to say four things: Thank you. I forgive you. I hope you can forgive me. I love you.
We should strive, George, to be constant contact with our Higher Power or at least several times per day. Some people have a prayer schedule and set aside 5 minutes very 4 hours like the Christian monks and the Muslims do. Us modern humans spend more time with our smart phones and other screens than we do giving our attention to the divine within us and we are the poorer for it.
Also, I suggest that you be careful praying to the universe as if it it were a vending machine like a Fairy Godmother, or Gene from a bottle who can dispense gifts.
The main purpose of prayer is to join with the cosmic consciousness, to become one with the Godhead, to heal the separation of ourselves from the Godhead and to go home. This can be achieved through meditation or what today is sometimes called mindfulness. This flow state, becoming one with the all can also be achieved through music, dance, athletics, communing with nature, great sex, etc.
So, George, prayer comes in many forms with the purpose of healing the separation of our ego from the all. One of the most lovely things in life is a prayer life based on the intention of bringing our will into alignment with God's will for us. The best prayer of all is the Our Father which Jesus suggested to us which combines all the best elements of a nourishing prayer life, gratitude, forgiveness, and setting aside of our egos so we can listen to the wishes of the universe for us.