I read online saying that to charge , I need at least 10% of the
battery current which is 0.4A. Is this right?
No, 0.4 A (0.1 'C' for a 4 Ah battery) is just the 'standard' rate. You can charge at a lower and it will simply take longer.
A 4 Ah battery charged at 0.4 A would theoretically take 10 hours to charge a from 'flat' to full, except the voltage must be kept at or below 7.2 V to prevent gassing, which happens after ~8 hours (by which time the battery is ~80% full). Then the charge current reduces exponentially towards a very low 'trickle' current, which takes another 8 hours or more to get the remaining 20% into the battery..
For battery longevity it is best to avoid discharging below 50% capacity. A 50% charged battery should take less than 4 hours to reach 80% capacity at 0.1 C. At 0.05 C (0.2 A in your case) it would take a bit less than 8 hours.
I got a 10W 17.5V 0.58A solar panel which I intend to charge a ... 6V
4AH battery.
There are two types of charge controller you can use - PWM (Pulse Width Modulation) and MPPT (Maximum Power Point Tracking).
PWM controllers simply connect the panel to the battery and rely on the panel to limit current. When voltage rises to the maximum permitted the controller pulses the current to reduce the average value. Any voltage difference between the rated panel voltage and battery voltage will be soaked up in the panel, wasting power. This is fine if the panel only puts out a few volts more than the battery needs (eg. 17.5 V down to 14.4 V for a 12 V battery) but is very inefficient if the battery voltage is a lot lower.
MPPT controllers work like a buck converter, pulsing power though an inductor which smooths out the current while dropping voltage with (almost) no loss. Unlike a normal buck converter an MPPT contoller monitors the input power and attempts to maximize it when possible. An MPPT controller can be over 90% efficient even when dropping the voltage by 50% or more. Since power = voltage x current that means the charging current can be higher than the panel current.
With an MPPT controller you could charge at 0.8 A and go from flat to 80% charge in ~5 hours. On a good sunny day that means you should be able to fully recharge the battery. With a PWM controller it might take 2 days or longer, but if the average power drain is low enough to never go below 50% charge (even on dull days when the charging current is lower) it should be OK.
