LONDON, Jan 24 (Reuters) - British services firms have seen another pick-up in growth this month, adding to signs of a modest recovery in the sluggish economy, although struggling factories are now being hit by the inflationary impact of tensions in the Red Sea, a survey showed.

Wednesday's snapshot of businesses is likely to reassure the Bank of England in the run up to next week's interest rate meeting that overall inflation pressure is easing, but may add to the case for moving only slowly towards cutting borrowing costs.

The preliminary S&P Global/CIPS UK Composite Purchasing Managers' Index (PMI), which spans services and manufacturing firms, rose to 52.5 in January, the highest in seven months and up from December's final reading of 52.1.

Economists polled by Reuters had forecast a slightly smaller increase to 52.2.

"Business activity and confidence are being in part driven by hopes of faster economic growth in 2024, in turn linked to the prospect of falling inflation and commensurately lower interest rates," Chris Williamson, S&P Global Market Intelligence's Chief Business Economist said.

"However, the surprising strength of growth in January, which has exceeded forecasts, may deter the Bank of England from cutting interest rates as soon as many are expecting, especially as supply disruptions in the Red Sea are reigniting inflation in the manufacturing sector," he added.

Among services firms, the PMI's headline measure climbed to 53.8, an eight-month high and up from December's 53.4.

Manufacturing continued to shrink although the pace of contraction slowed a bit to 47.3 from 46.2, its closest to the no change level of 50.0 since last April.

Factories reported the first growth in input costs since April as the re-routing of ships away from the Red Sea pushed up freight costs and delivery times rose for the first time in a year. But an increase in prices charged by manufacturers was only modest.

The pace of cost increases for services firms - chiefly wages - grew by the least in three months.

Prices charged by firms overall increased at their weakest pace since October 2023.

There were other positive signs in the "flash" January PMI including the first growth in employment in five months, the strongest increase in new work since last May and the most optimism about the outlook also since last May. (Reporting by William Schomberg; Editing by Christina Fincher)