So "intense time of prayer, singing, praise, and learning more about and focusing on Jesus Christ" is not worship, in my opinion. I know people call it that now, colloquially, but that is a very modern idea. The description is closer to veneration.
The reason I said it was gnostic is because what you've described is particularly intellectual, it's about the mind. This is *very* much a feature of gnosticism.
That description of worship is nowhere in the OT, nowhere in the practice of Israel, nowhere in the NT, and is basically nowhere (other than gnosticism) in the history of Christianity until fairly recently.
Corporate worship is when the people of God assemble, uniquely to participate in a ritual meal with their God. Singing, praying, bowing down is all part of this, to be sure. And distractions aren't part of it and aren't welcome - this is why it's silly to draw an equivalence between a child being a child and crying, or a mother tending to her child's needs, and an adult actively not participating by doing something else in spite of being physically present. Children participate as children, mother as mothers, you as you, me as me.
Being together is literally what Church means. The word is ecclessia, assembly, those called out (to assemble). Your children are a part of the assembly if they are baptized - and they should be. So "when you come together as a church" as St Paul says - you should actually come together. It is no accident that this coming together as a church is related to the Eucharist, and why Christian gatherings everywhere and always until the Reformation were associated with the Eucharist, and why that worship was a meal.
Christ is the focus at that time. And we are the Body of Christ, together we are His hands and feet, we make Christ present in the world. If a child is struggling, love them. Pick them up. These people are your family, would you ask your daughter to leave the family gathering if she's crying? Expect them to go outside if they're "distracting" you? Those children are His children (cf Ezekiel 16:20-21) and He says to bring them to Him (Matthew 19:14).