As pointed out, the Office of Readings, and Morning Prayer, are two separate offices. There is a rubric for joining them together, but let’s look at the structure of each independently, then I’ll show how to blend them.
OOR
If first Office of the day: “Lord open my lips…”
Invitatory Antiphon
Invitatory Psalm
If not the first office of the day, e.g. said the evening before: “Lord, come to my assistance…”
Hymn
Three psalms or sections of psalms
Verse
Biblical reading
Responsory
Patristic/Hagiographic reading
Responsory
Te Deum (Sundays, feasts and solemnities except Sundays in Lent)
Collect (Seasonal, or of the saint/feast)
Conclusion
MP
If first Office of the day: “Lord open my lips…”
Invitatory Antiphon (optional)
Invitatory Psalm (optional)
If not the first office of the day, e.g. you said the OOR earlier: “Lord, come to my assistance…”
Hymn
First psalm
Old Testament canticle
Second Psalm
Short reading
Responsory
Gospel canticle antiphon
Gospel Canticle (Benedictus)
Intercessions
Our Father
Collect (of the day, season, or saint)
Conclusion
To join the two Offices together
“Lord open my lips…”
Invitatory Antiphon
Invitatory Psalm
Hymn (either the one specified for Morning Prayer or the one for the Office of Readings)
Three psalms or sections of psalms from the Office of Readings
Verse
Biblical reading
Responsory
Patristic/Hagiographic reading
Responsory
Te Deum (Sundays, feasts and solemnities except Sundays in Lent)
First psalm from Morning Prayer
Old Testament canticle
Second Psalm
Short reading
Responsory
Gospel canticle antiphon
Gospel Canticle (Benedictus)
Intercessions
Our Father
Collect (of the day, season, or saint)
Conclusion
In other words, you get to eliminate one hymn, and you only say the Collect of Morning Prayer.
As always I advise people wanting to get into the LOTH to spend some quality time with the General Instruction of the Liturgy of the Hours (if you don’t have the books you can use Google to find it), and the Ordinary section in the books. These are invaluable for correctly praying the Office if not using a website.
PS in principle, you can also join the Office of Readings to either Mid-day prayer, Evening Prayer or Night Prayer. The formula is the same as if for joining Morning Prayer, but without the Invitatory verse/antiphon/psalm.
Edit: corrected a copy/paste error for the joint Offices, I had two hymns when only one is required just after the Invitatory.