"I thought I wanted it. But the burden is a heavy one. It's too heavy. If you wish me to bear it then defend me. And my children."

I wonder what burden she feels she's holding at this moment. We know that she isn't speaking of the burden of ruling and managing the realm for the last several years while her father was sick, because while she was on Dragonstone, those duties fell to the Hand and the Queen. As far as the show tells us, Rhaenyra has no specific duties she's fulfilling as heir, nor has she been doing anything in particular to prepare to rule. We're not shown that she's doing anything with her time beyond anything beyond managing Dragonstone with Daemon and growing her family. So what exactly in her life at this moment is she seeing as this massive burden that she bears because she's heir to the throne?

I think the burden she's speaking of here is actually the pressure she's feeling as she's finally being faced with the idea that her father won't always be around to help her avoid the consequences of her actions. So even though it's obvious that he's on the edge of death, in immense pain, barely hanging on, and despite the fact that she hasn't visited him at this point in literal years despite knowing about his ailing health and she's done nothing to support him in ruling in his last years, she still feels entitled enough to demand that he do another massive favor for her. At this point, Rhaenyra has been enabled by her father for so many years that she doesn't know how to solve a problem without him. So she plays the heir card on Viserys to get him to defend her one more time: if you want me to be queen, and you don't want me to be held accountable and face consequences, help me get away with it one more time.

Rhaenyra is upset not because of any real burden she has as heir but because she feels like the walls are closing in and she's under attack. I would feel some sympathy for her if it weren't for the fact that she wouldn't be in this situation if she'd made better choices. If literally anyone else in the realm had obvious bastards they were using to usurp a seat of power, there would be huge consequences. Rhaenyra knows this and is doing it anyway because she thinks her position as heir to the throne gives her the power to be able to do it and get away with it. Her birthright as a Targaryen says she deserves absolute power because she's better than other people, so she can do no wrong and nobody can say otherwise to her. She believes she's clever enough to pass the deception, and otherwise her father can punish anyone who speaks up... until Vaemond goes to court, and her father is not there to blankly support her no matter what. Now, she's faced with people who intend to uphold the law of the land and she's upset that she might actually be confronted with her problems head on and not have someone to shield her this time.