This morning I was going through a Unicom JLPT 1 kyuu kanji study book (an older version of this one). It had a list of six jukugo using 修 but I was struck by the wide variety of meanings. Here they are with some examples from the book or my Casio XD-GW9600:

Kanji Study: shuu

修学 しゅうがく learning; schooling:
修学旅行 school excursion; field trip
修飾 しゅうしょく decoration:
can be used rhetorically その話にはかなり修飾がある。 That story is highly embellished.
修正 しゅうせい correction; revision:
間違いを修正する。 Correct one’s mistake.
修繕 しゅうぜん repair; mending:
お待ちの間に修繕いたします。 Repairs done while you wait.
改修 かいしゅう improvement:
この橋は20年間改修されていない。 This bridge has not been repaired for twenty years.
専修  せんしゅう specialization:
英語専修 a special English course.

I should also include one of my favorite words to the list: 修行 しゅぎょう training (like for becoming a super ninja) Note it has a short しゅ without the trailing う. All the above jukugo have the longer しゅう. Interestingly there is a record in EDICT for it spelled as しゅうぎょう but I think that must be a mistake.

The Kanji Learner’s Dictionary gives 修 the meaning of ‘cultivate’ or ‘repair.’ I can see those basic meanings in the above examples, but I think this is an illustration of why it is important to learn words or example sentences instead of simply relying on a set English meaning for each individual kanji.
In my opinion systems like Heisig’s are hugely popular with beginners because it works with lower level kanji well. Most beginner level kanji seem to have a singular and well defined meaning. 青 is blue; 牛 is cow. Heisig isn’t so popular with advanced learners because it becomes increasingly difficult to pin down a single English meaning for the more abstract advanced level kanji.
As for my kanji studies… I’m still 修行中.  To be continued…