Those who have faith in Christ are adopted into his family for eternity, and they can now have a relationship with God. Prayer is something we can do at any time, and God listens because Jesus has paid the price to restore that relationship.
So God is always present to listen to our prayers. God is always loving us. God is sovereign over everything. In so many ways God is near to us. Phrases like "the presence of God" meaning God only being present at certain times, such as only in prayer or only when we are doing something truly noble, may be harmful. The people who listen to or say these things may start to think or feel that they must meet a quota of good deeds to feel like God loves them or is present in their circumstances.
When we are not praying or doing very noble things, it's possible we may feel it harder to love God as much or see his beauty quit as clear at that moment. But that is about us, and not God being any farther away from us. As humans, we're sometimes limited on how many things we can focus on in one given moment.
We should rejoice in what Christ has allowed for us and for how we are always loved. We shouldn't try to feel like we'll gain motivation to do good deeds if we say good deeds will get us into some new state with or closer to God. We are already as close as we can be. We can grow in more knowledge, and therefore that may help us grow in love for God (more things to know to marvel at or praise him for)--but that is about us moving closer to him, not him moving closer to us.
Motivation to do good deeds can happen when we really believe that God is such a gracious God. Christ has saved us and anyone who is willing to come to him in faith. A natural response to being saved is to be grateful, and our appreciation and love for the person who saved us causes us to want to serve or honor that great person.