Note: ては may contract to ちゃ (てはかなわない → ちゃかなわない) in casual speech.
The grammar pattern ~てはかなわない expresses that a situation is unbearable, intolerable, or too much to handle. It conveys the speaker's strong feeling that they cannot stand something or that it would be unbearable if something were to happen. The かなわない here is the negative form of かなう (to be bearable/tolerable), so the literal sense is "if ~ happens, it is not bearable."
This pattern is used to complain about or express frustration with situations that cause discomfort, inconvenience, or distress. It is a fairly literary or emphatic expression and is characteristic of N1-level Japanese.
Attach てはかなわない to the て-form of verbs, い-adjectives (くて), or な-adjectives (で):
Verb: V[て] + はかなわない
い-adjective: Adj[くて] + はかなわない
な-adjective: Adj + では かなわない
The pattern often implies that the speaker is personally affected and is expressing a complaint or grievance. It sounds stronger and more emphatic than simply saying こまる困る or たまらない. It is closely related to ~てはたまらない and ~てはやりきれない, which share similar meanings of intolerability, though ~てはかなわない can carry a slightly more indignant or exasperated tone.
In casual speech, ては is often contracted to ちゃ (or じゃ after で-form):
It frequently appears with expressions like こう (like this), まいにち毎日 (every day), いつも (always), or conditional/hypothetical situations to emphasize the ongoing or repeated nature of the unbearable condition.