First, usage notes:
1. Fledgling refers to a beginner in an activity, but it also has connotations of youthful inexperience. It is a great word, but it is not used very often for adults. "Beginner" or "novice" is more common when talking about language learning.
2. To elucidate is a high register (advanced, more formal or technical) word that is the equivalent of "to explain" or "to make clear" in neutral/informal speaking and writing. Elucidation is also high register and is the equivalent of ]"clarification" or "explanation".
3. At your convenience is a common phrase that shows politeness and willingness to get along. It is useful both for informal and formal situations, like business or education. It is similar to "when you get a chance", which is more informal.
In short, elucidate, elucidation, and at your convenience are good to use. You can call yourself a fledgling but I wouldn't use the word to describe another person.
Sentence corrections:
"At that time, I was a fledgling/beginner/novice at English, but now I speak English perfectly."
*"My/the economics teacher was inexperienced in her subject, so I left her class."
"I don't understand; could you elucidate/explain/clarify/make it clearer?"
"I don't need any elucidation/explanation/clarification from you about what happened on that day."
"You can text me back at your convenience. No problem!"
"You can come to my house at your convenience." or "You can come over anytime you want."
*Fledgling here would be considered an insult.