can you sue a tenant in georgia for additional money not covered by there security deposit for damages n clean up done to an apt

Yes, for the cost to clean or repair the property.

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The short answer is absolutely yes. The lease probably spells out the specific contractual obligations you have to return the property (broom swept clean is an industry term). The security deposit MAY have been written up as liquidated damages but this is unlikely, as the landlord would not limit themselves that way in advance.

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