The -며 verb form indicates doing one thing at the same time as another (-면서 is similar but more colloquial).
e.g. 동생은 친구와 놀겠다며(놀겠다고 말하며) 나갔다 = My brother went out saying he wanted to play with his friend.
If you end a phrase with -며, it gives an open-ended nuance that something else is happening as well.
So,
1. 싱그러운 여름이 되길 바란다 - I wish you a pleasant summer (closed full sentence - expresses an idea)
2. 싱그러운 여름이 되길 바라며 - Wishing you a pleasant summer (incomplete and open, like a backdrop of another thing)
Phrases ending with -며 are often used as a title or heading, closing of a letter, etc.
For example, the play "Waiting for Godot" is titled 고도를 기다리며 in Korean.