HARTFORD — Connecticut lawmakers will return to the state Capitol on Tuesday to finish work on a couple of remaining budget-related bills that touch on everything from minimum staffing levels for state troopers to state ­financial assistance for an ­expanded number of businesses.

But the big focus of the special legislative session is a proposal to impose new state taxes on roll-your-own cigarette stores. The proposal, which would generate an estimated $3.4 million in state tax revenue, affecting 15 to 20 shops, is at the center of a campaign fund-raising controversy involving House Speaker Chris Donovan.

Donovan, a Meriden Democrat, is running for the Fifth Congressional District. His ­financial director, Robert Braddock Jr., was recently ­arrested by federal authorities and charged with allegedly conspiring to hide the source of $20,000 in contributions, linked to the roll-your-own bill, to the congressional campaign. Braddock’s attorney says his client is innocent.

While he plans to vote, Donovan will not preside over Tuesday’s session because of the controversy. Instead, House majority leader ­Brendan Sharkey of Hamden will handle the special session in the House.

Both the House and Senate are expected to take up two bills that spell out many ­details of the revised $20.1 billion budget that was passed this year.

Included in a draft of the proposed legislation is a provision to no longer require that the State Police maintain 1,248 troopers. Instead, it calls for the commissioner to appoint ‘‘a sufficient number of sworn State Police personnel to efficiently maintain the operation of the division.’’

One draft bill also revives legislation extending state assist­ance to more small businesses in an effort to boost job growth.

The Republican leader in the House, Lawrence Cafero Jr., accused the majority Democrats of loading up the legislation with pet projects. He said there are more than 100 bills represented in the two budget-related bills. That is in contrast to the 344 bills ­approved during the regular legislative session, which ended last month.