Coalescence refers to a process in which two particles become one. In linguistics this is used to describe the phonological process which turns two sounds into one. For example, the -ng in sing is pronounced as a single sound.
Though most English speakers pronounce the final two letters of sing as one sound, speakers in certain parts of England pronounce them as two. For example, this is the case for speakers in Manchester, Liverpool and Birmingham. For them coalescence doesn't apply.
For those English speakers who pronounce the -ng of sing as a single sound, this is also true for the -ng in singer. Likewise, for those who pronounce the -ng of sing as two sounds, this is also the case in singer. However, all English speakers pronounce the -ng of finger as two sounds.
For most English speakers, the -ng of finger and singer are pronounced differently. The difference is a subtle one, though, so they may not be aware of it. For speakers who don't have coalescence of the -ng in singer, the words finger and singer have the same -ng sound.
Most English speakers apply coalescence in the pronunciation of words such as sing and singer. The letters -ng represent just one sound. In the English of certain areas of England, though, these letters are pronounced as two separate sounds. For these speakers the words finger and singer rhyme more completely.