Common Breathlessness, Rash, Vomiting, Angioedema (swelling of deeper layers of skin), Nausea, Fatigue, Increased liver enzymes, Edema, Diarrhoea, Decreased blood pressure, Flushing (sense of warmth in the face, ears, neck and trunk), Decreased albumin level in blood, Increased glucose level in blood, Urticaria.

Expert advice for Asginase Injection

Asparaginase can also make you flush, breathe faster, or have a fruit-like breath. This is normal and not a cause for concern.
Do not receive a live vaccine (eg, measles, mumps) while you are taking asparaginase.
Do not take this drug if you are allergic to asparaginase or any of its ingredients.
Seek immediate medical attention if you feel confused, drowsy, have excessive thirst or observe an increased frequency or volume of urination. 
If you are a patient with diabetes, monitor your blood sugar levels closely.
Ensure that you undergo periodic laboratory tests to monitor response to the therapy and any side effects.