Possible Clues:

• A sucker may stick around being sick
• Medication taken by the sick in a glass container
• Sweet paste - get sick swallowing it? On the contrary!
• That sucker will make one sick in 30 down
• Ate pills? One medicated sweet is the alternative
• If you're sick, stick around like a sucker
• Still a sucker with that ape around
• Lozenge, small sweet for sucking
• I will shortly get in the sticky stuff for a sucker
• Medication of pills taken with tea